We begin by discussing why we should care about Clifford Algebra. (If you want an overview of how Clifford Algebra actually works, skip to section 2.)
The cross product only makes sense in three dimensions.  The wedge product is well behaved in any number of dimensions, from zero on up. 
The cross product is defined in terms of a “right hand rule”.  A wedge product is defined without any notion of handedness, without any notion of chirality. This is discussed in more detail in section 2.13. This is more important than it might seem, because it changes how we perceive the apparent symmetry of the laws of physics, as discussed in reference 3. 
The cross product only applies when multiplying one vector by another.  The wedge product can multiply any combination of scalars, vectors, or highergrade objects. 
The wedge product of two vectors is antisymmetric, and involves the sine of the angle between two vectors ... and the same can be said of the cross product.
∇ F = 
 J (1) 
It is worth learning Clifford algebra just to see this equation. For details, see reference 4.
Also: In their traditional form, the Maxwell equations seem to be not left/right symmetric, because they involve cross products. However, we believe that classical electromagnetism does have a left/right symmetry. By rewriting the laws using geometric products, as in equation 1, it becomes obvious that no righthand rule is needed. A particularly pronounced example of this is Pierre’s puzzle, as discussed in reference 3.
Similarly: The traditional form of the Maxwell equations is not manifestly invariant with respect to special relativity, because it involves a particular observer’s time and space coordinates. However, we believe the underlying physical laws are relativistically invariant. Rewriting the laws using geometric products makes this invariance manifest, as in equation 1.
As an elegant application of the basic idea that the electromagnetic field is a bivector, reference 5 explains why a field that is purely an electric field in one reference frame must be a combination of electric and magnetic fields when observed in another frame.
As a more mathematical application of equation 1, reference 6 calculates the field surrounding a long straight wire.
The purpose of this section is to provide a simple introduction to Clifford algebra, also known as geometric algebra. I assume that you have at least some prior exposure to the idea of vectors and scalars. (You do not need to know anything about matrices.)
For a discussion of why Clifford algebra is useful, see section 1.
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In addition to scalars and vectors, we will find it useful to consider moregeneral objects, including bivectors, trivectors, et cetera. Each of these objects has a clear geometric interpretation, as summarized in figure 1.
That is, a scalar can be visualized as an ideal point in space, which has no geometric extent. A vector can be visualized as line segment, which has length and orientation. A bivector can be visualized as a patch of flat surface, which has area and orientation. Continuing down this road, a trivector can be visualized as a piece of threedimensional space, which has a volume and an orientation.
Each such object has a grade, according to how many dimensions are involved in its geometric extent. Therefore we say Clifford algebra is a graded algebra. The situation is summarized in the following table.
object  visualized as  geometric extent  grade  
scalar  point  no geometric extent  0  
vector  line segment  extent in 1 direction  1  
bivector  patch of surface  extent in 2 directions  2  
trivector  piece of space  extent in 3 directions  3  
etc.  
For any vector V you can visualize 2V as being twice as much length, and for any bivector B you can visualize 2B as having twice as much area. Alas this system of geometric visualization breaks down for scalars; geometrically all scalars “look” equally pointlike. Perhaps for a scalar s you can visualize 2s as being twice as hot, or something like that.
For another important visualization idea, see section 2.12.
As mentioned in section 1 and section 6, bivectors make cross products obsolete; any math or physics you could have done using cross products can be done more easily and more logically using a wedge product instead.
The scalars in Clifford algebra are the familiar real numbers. They obey the familiar laws of addition, subtraction, multiplication, et cetera. Addition of scalars is associative and commutative. Multiplication of scalars is associative and commutative, and distributes over addition.
The vectors in Clifford algebra can be added to each other, and can be multiplied by scalars in the usual way. Addition of vectors is associative and commutative. Multiplication by scalars distributes over vector addition. We will introduce multiplication of vectors in section 2.5.
Presumably you are familiar with the idea of adding scalars to scalars, and adding vectors to vectors (tip to tail). We now introduce the idea that any element of the Clifford algebra can be added to any other. This includes adding scalars to vectors, adding vectors to bivectors, and every other combination. So it would not be unusual to find an element C such that:
C = s + V + B (2) 
where s is a scalar, V is a vector, and B is a bivector.
This clearly sets Clifford algebra apart from ordinary algebra.
Remark: Sometimes nonexperts find this disturbing. Adding scalars to vectors is like adding apples to oranges. Well, so be it: people add apples to oranges all the time; it’s called fruit salad. In contrast, it is proverbially unwise to compare apples to oranges, and indeed we will not be comparing scalars to vectors. Addition (s + V) is allowed; comparison (s < V) is not.
Terminology: The most general element of the Clifford algebra we will call a clif. In the literature, the same concept is called a multivector, but we avoid that term because it is misleading, for reasons discussed near the end of section 2.9.
Presumably you already know how to add vectors graphically, by placing them tiptotail as shown in figure 2. By extending this idea, we can also add bivectors graphically, by placing them edgetoedge as shown in figure 3.
We add bivectors edgetoedge, in analogy to the way we add ordinary vectors tiptotail. In this example, edge b adds tiptotail to edge x to form the top edge of the sum. Similarly, edge z adds tiptotail to edge d to form the bottom edge of the sum. Edge c cancels^{1} edge w since they are equal and opposite. Edges a and y survive unchanged to become the vertical edges of the sum.
As a concrete example of addition of bivectors, consider a gyroscopic precession problem, as follows: The green bivector is the initial angular momentum of the system, and the small purple bivector is torque∧time. Then the yellow bivector is the new angular momentum, which has a new orientation due to precession.
The formula for angular momentum is given in section 3.
Given any clif C, we can talk about the grade0 piece of it, the grade1 piece of it, et cetera.
Notation: The gradeN piece of C is denoted ⟨C⟩_{N}.
We will often be particularly interested in the scalar piece, ⟨C⟩_{0}.
Note: If you are familiar with complex numbers, you can understand the ⟨⋯⟩ operator as analogous to the ℜ() and ℑ() operators that select the real and imaginary parts. However, there is one difference: You might think that the imaginary part of a complex number would be imaginary (or zero), but that’s not how it is defined. According to longestablished convention, ℑ(z) is a real number, for any complex z. Clifford algebra is more logical: The vector part of any clif is a vector (or zero), the bivector part of any clif is a bivector (or zero), et cetera. See reference 1 for more on this.
Axiom: We postulate that there is a geometric product operator that can be used to multiply any element of the Clifford algebra by any other.
Notation: The geometric product of A and B is written AB. That is, we simply juxtapose the multiplicands, without using any operator symbol.
As we shall see in section 2.6, the geometric product AB is distinct from the dot product A·B and also from the wedge product A∧B.
We further postulate that the geometric product is associative and distributes over addition:
 (3) 
where A, B, and C are any clifs.
Multiplication is not commutative in general, as we shall see in section 2.6 and elsewhere, but we remark that the special case of multiplying by scalars is commutative:
We have already asserted that any clif can be multiplied by any other clif. Multiplying a vector by a vector is a particularly interesting case. At this point Clifford algebra makes a dramatic departure from ordinary vector algebra.
Given two vectors P and Q, we know that the geometric product PQ exists, but (so far) that’s about all we know. However, based on this mere existence, plus what we already know about addition and subtraction, we can define two new products, namely the dot product P·Q and the wedge product P∧Q, as follows:

Equation 4a is a very useful formula, but you should not become overly attached to it, because it only applies to grade=1 vectors. It doesn’t work for higher or lower grades. See section 2.10 for a more general formula.
Equation 4b is only slightly more general. It works for any combination of vectors and/or scalars (grade≤1). See section 2.11 for some more general formulas.
As an immediate corollary of equation 4, we can reexpress the geometric product of two vectors as:
 (5) 
This useful formula only works when both P and Q are vectors. I keep mentioning this, because some authors take equation 5 to be the “definition” of geometric product (based on some sort of preexisting notion of dot and wedge). They can get away with that for the product of two plain old vectors, but it fails for higher (or lower) grades, and creates lots of confusion.
As another corollary of equation 4, we see that the dot product of two vectors is symmetric, while the wedge product of two vectors is antisymmetric:

Terminology: The wedge product is sometimes called the exterior product. (This is not to be confused with the tensor product ⊗, which is sometimes called the outer product.)
In this document we prefer to call it the wedge product. We shall not be concerned with derivatives of any kind, exterior or otherwise.
Beware: Reference 8 introduces the term “antisymmetric outer product” to refer to the wedge product, and then shorthands it as “outer product”, which is highly ambiguous and nonstandard. It conflicts with the longstanding notion of outer product, as discussed in section 7.
Let’s investigate the properties of the dot product. We restrict attention to ordinary grade=1 vectors. We will show that the dot product defined here behaves just like the dot product you recall from ordinary vector algebra. For starters, we are going to argue that P·P behaves like a scalar.
One characteristic behavior of a scalar (in the geometric sense) is that if you rotate it, nothing happens. This is very unlike a vector, which changes if you rotate it (unless the plane of rotation is perpendicular to the vector).
As an introductory special case, consider rotating P by 180 degrees, assuming P lies in the plane of rotation. That’s easy to do: a 180 degree rotation transforms P into −P. We are pleased to see that this transformation leaves P·P unchanged. This is easy to prove, using the definition (equation 4) and using the fact that multiplication by scalars is commutative: Just factor out two factors of 1 from the product (−P)·(−P).
It’s also obvious that a rotation in a plane perpendicular to P leaves P·P unchanged, which is reassuring, although it doesn’t help distinguish scalars from anything else.
Tangential remark: More generally, we assert without proof that P·P is in fact invariant under any rotation (i.e. any amount of rotation in any plane). We are not ready to prove this, since we haven’t yet formally defined what we mean by rotation ... but we will pretty much insist that rotation leave P·P invariant, because we want P·P to be a scalar, and we want √(P·P) to be the length of P, and we want rotations to be lengthpreserving transformations. (This isn’t a proof, but it is an argument for plausibility and selfconsistency.)
If P·P is a scalar, it is easy to show that P·Q is a scalar, for any vectors P and Q. Just define R := P + Q and then take the dot product of each side with itself:
R·R = P·P + 2 P·Q + Q·Q (7) 
where every term except 2 P·Q is manifestly a scalar, so the remaining term must be a scalar as well.
This leaves us pretty much convinced that the dot product between any two vectors is a scalar. It can’t be a vector or anything else we know about. To get here, we didn’t do much more than postulate the existence of the geometric product, and then do a bunch of arithmetic.
Terminology: If vector Q is equal to P, or is equal to P multiplied by any nonzero scalar, we say that P and Q are parallel and we symbolize this as P ∥ Q.
Based on what we already know (mainly the symmetry properties, equation 6) we can deduce that if P and Q are parallel, then P∧Q = 0 and PQ = P·Q; that is:
PQ = QP = P·Q iff P ∥ Q (8) 
which gives us a useful test for detecting parallel vectors.
Terminology: If P·Q = 0, we say that vectors P and Q are perpendicular or equivalently orthogonal and we symbolize this as P ⊥ Q.
If vectors P and Q are orthogonal, then P·Q = 0 and PQ = P∧Q; that is:
PQ = −QP = P∧Q iff P ⊥ Q (9) 
In general, in the case where P and Q are not necessarily parallel or perpendicular, the geometric product will have two terms, in accordance with equation 5.
Lemma: We can resolve any vector P into a component P_{Q} which is parallel to vector Q, plus another component (P−P_{Q}) which is perpendicular to Q. Proof by construction:
P_{Q} := Q 
 (10) 
This lemma is conceptually valuable, and frequently useful in practice. (See e.g. section 2.18.)
P_{Q} is called the projection of P onto the direction of Q, or the projection of P in the Qdirection.
It is an easy exercise to show the following:
 (11) 
Now, let’s investigate the properties of the wedge product of two vectors. We anticipate that it will be a bivector (or zero). We use the same line of reasoning as in section 2.7. We begin by considering the case where P∧Q is nonzero.
You can easily show that the wedge product P∧Q is invariant with respect to 180 degree rotation in the PQ plane. That is, just replace P by −P and Q by −Q and observe that nothing happens to the wedge product. This tells us the product is not a vector in the PQ plane. We remark without proof that this result is invariant under any rotation (however small or large) in the PQ plane.
Things get more interesting if we have more than two dimensions, because that allows us to investigate additional planes of rotation.
To make things easy to visualize, let us replace Q by Q′, where Q′ is the projection of Q in the directions perpendicular to P. We can always do this, using the methods discussed in section 2.8. According to equation 11, we know P∧Q′ is equal to P∧Q.
Choose any vector R perpendicular to both P and Q′. Rotate both vectors in the PR plane by 180 degrees. This transforms P into −P, but leaves Q′ unchanged (since it is perpendicular to the plane of rotation). That means the rotation flips sign of the wedge product, P∧Q.
Similarly a rotation in the QR plane flips the sign of the wedge product. As a final check, we perform an inversion, i.e. the operation that transforms any vector V into −V, not limited to any plane of rotation. This leaves the wedge product unchanged.
Taking all these observations together, we find that P∧Q behaves exactly as we would expect a bivector to behave, based on the description given in section 2.1: a patch of surface with a direction of circulation around its edge. That is: the area is unchanged if we rotate things in the plane of the surface, but if we rotate things 180 degrees in a plane perpendicular to the surface, the surface flips over, reversing the sense of circulation.
The idea of wedge product generalizes to more than two vectors. For example, with three vectors, we generalize equation 4 as follows:
P∧Q∧R := 
 (PQR + QRP + RPQ − RQP − QPR − PRQ) (12) 
You can skip the following equation if you’re not interested, but if you want the fully general expression, it is:
q_{1}∧q_{2}∧q_{3}⋯q_{r} := 

 sign(π) q_{π(1)} q_{π(2)} q_{π(3)}⋯ q_{π(r)} (13) 
where the sum runs over all possible permutations π. There are r! such permutations, and sign(π) is defined to be +1 for even permutations and −1 for odd permutations. This will be an object of grade r if all the vectors q_{1}⋯q_{r} are linearly independent; otherwise it will be zero.
People like to say “the wedge product is antisymmetric” ... but you have to be careful. It is antisymmetric with respect to interchange of any two vectors ... not any two clifs. For example:
 (14) 
for any scalar s and any clif C.
Terminology: A blade is defined to be any scalar, any vector, or the wedge product of any number of vectors.
Terminology: Any clif that has a definite grade is called homogeneous. It is necessarily either a blade or the sum of blades, all of the same grade.
Example: The sum s + V (where s is a scalar and V is a vector) is not homogenous. It does not have any definite grade. It is certainly not a blade.
Example: In four dimensions, the quantity γ_{0} γ_{1} + γ_{2} γ_{3} is homogeneous but is not a blade. It has grade=2, but cannot be written as just the wedge product between two vectors.
Terminology: As previously mentioned, we use the term clif to cover the most general element of the Clifford algebra. In the literature, the same concept is called a multivector, but we avoid that term because it is misleading. The problem may be due in part to the etymology suggested by the sequence:
vector, bivector, trivector, ⋯, multivector (WRONG) (15) 
in contrast to the correct sequence:
vector, bivector, trivector, ⋯, blade (RIGHT) (16) 
Terminology: Do not confuse a trivector with a 3vector. A trivector is visualized as the 3dimensional region spanned by three vectors. This region may be embedded in a space that is 3dimensional or higher. In contrast, a 3vector is a single vector that lives in a space with exactly 3 dimensions.
Terminology: To describe the grade of a blade, the recommended approach is to mention the grade explicitly. For example, we say a bivector has grade=2, while a trivector has grade=3, and so on.
There is another, nonrecommended approach, in which a bivector is called a 2blade, while a trivector is called a 3blade, and so on. This is risky because of possible confusion as to whether the number refers to grade or dimension. Note that a 3blade has grade=3, while a 3vector has dimension=3, so confusion is to be expected.
We have already defined the wedge product of arbitrarily many vectors, according to equation 13.
We now define the wedge product between any blade and any blade. The rule is simple: unpack each blade as a wedge product, remove the parentheses, and apply equation 13:
 (17) 
where the RHS is defined by equation 13. As an obvious corollary of this definition, the wedge product has the associative property.
It is easy to see that for any two blades P and Q, which are of grade p and q respectively, the grade of P∧Q will be p+q (unless the product happens to be zero, in which case its grade is zero).
We can use that idea in the other direction, as follows: As discussed in section 2.14, the full geometric product PQ is liable to contain terms of all grades from p−q to p+q inclusive (counting by twos). The wedge product consists of just those terms with the highest possible grade. In symbols:
 (18) 
Given this definition of blade wedge blade, we can generalize to any clif wedge clif, simply by saying the wedge product distributes over addition:
V∧(A + B) = V∧A + V∧B (19) 
We hereby define the dot product of two blades to be the lowestgrade part of the geometric product. That is, if P has grade p and Q has grade q, then the dot product will have grade p−q. In symbols:
 (20) 
Just as the wedge product was the topgrade part of the geometric product, the dot product is the bottomgrade part.
Let’s be clear: The definition of dot product depends more on its grade than on its symmetry. The dot product of two vectors is symmetric, while the dot product of a vector with a bivector is antisymmetric:
 (21) 
You can check that this moregeneral definition of dot product is consistent with what we said back in section 2.6 about the dot product of vectors.
Given this definition of blade dot blade, we can generalize to any clif dot clif, simply by saying that the dot product distributes over addition. That is,
V·(A + B) = V·A + V·B (22) 
There is a very interesting way to visualize the wedge product. Consider the product C∧V, where C is a clif of any grade and V is a vector. The idea is to use C as a paintbrush, dragging C along V. The dragging motion is specified by the direction and magnitude of V. We keep C parallel to itself during the process. For instance, in figure 1 or figure 4, we form the parallelogram P∧Q by dragging the vector P along Q. Similarly in figure 1 we form the parallelepiped P∧Q∧R by dragging the parallelogram P∧Q along R.
The paintbrush picture is a little dodgy in the case where C is a scalar, but we can repair it by rewriting s∧V as 1∧(sV), for any scalar s. That is, we take the scalar 1 (which is pointlike) and drag it for a distance sV in the V direction. It just paints a copy of sV.
The orientation of the bivector P∧Q can be thought of as a “direction of circulation” marked on the parallelogram, namely moving in the P direction then moving in the Q direction. In figure 4, Q∧P = − P∧Q because they have the opposite direction of circulation. (They have the same magnitude, just opposite orientation.)
Chirality is a fancy word for handedness. It describes a situation where we can define the difference between righthanded and lefthanded. The foundations of Clifford algebra do not require any notion of handedness. This is important, because many of the fundamental laws of physics are invariant with respect to reflection, and Clifford algebra allows us to write these laws in a way that makes manifest this invariance.
The exception is the Hodge dual, which requires a notion of handedness, as discussed in section 5. This can be considered an optional feature, added onto the basic Clifford algebra package.
The following three concepts are all optional, and are all equivalent: (1) A notion of “front” versus “back” side of the bivector; (2) a notion of chirality such as the “righthand rule”; and (3) a notion of “clockwise” circulation.
We emphasize that for most purposes, we do not need to define any of these three concepts. We are just saying that if you did define them, they would all be equivalent. Except in section 5, we are not going to rely on any notion of clockwise or righthandedness or frontversusback. Instead, we rely on orientation as specified by circulation around the edge of the bivector, which is completely geometrical and completely nonchiral.
We make a point of keeping things nonchiral, to the extent possible, because it tells us something about the symmetry of the fundamental laws of physics, as discussed in reference 3.
In general, if you multiply an object of grade r by an object of grade s, the geometric product is liable to contain terms of all grades from r−s to r+s, counting by twos, as we see in the following:
Example: Let {γ_{1}, γ_{2}, γ_{3}, γ_{4}} be a set of orthonormal spacelike vectors, as discussed in section 2.18, and define:
 (23) 
Both A and B are homogeneous of grade 2. Indeed they are 2blades. It is easy to calculate the geometric product:
 (24) 
and therefore

For highgrade clifs A and B, the geometric product AB generally leaves us with a lot of terms. As discussed in section 2.11 the bottomgrade term is the dot product (e.g. equation 25a). Meanwhile, as discussed in section 2.10, the topgrade term is the wedge product (e.g. equation 25c). Alas, there are no special names for the other terms, such as the middle term in the previous example (e.g. equation 25b).
If we temporarily restrict attention to plain old vectors (grade=1 only), we reach the remarkable conclusion that the geometric product of two vectors has only two terms, a scalar term and a bivector term (assuming the bivector term is nonzero):
 (26) 
which we can compare and contrast with equation 4, which we reproduce here:
 (27) 
still assuming V and W are vectors.
Sometimes one sees introductory discussions that start by presenting the properties of the dot product and wedge product, and then “define” the geometric product as
 (28) 
which turns the discussion on its head relative to what we have done here. We started by postulating the existence of the geometric product, and then used it to work out the properties of dot and wedge.
They can get away with equation 28 when talking about vectors, but it doesn’t generalize well to objects of any grade higher or lower than 1, and it gets students started off on the wrong foot conceptually. Here is a simple counterexample:
 (29) 
where s is any scalar and C is any clif. Here is another important example:
 (30) 
In ordinary Euclidean space, whenever you compute the dot product of a vector with itself, the result is positive; that is, S·S > 0. In general, whenever this dot product is positive, we say that the vector S is spacelike.
In special relativity, i.e. in Minkowski space, we find that some vectors have the property that T·T < 0. In that case, we say that the vector T is timelike.
In a space where spacelike and timelike vectors exist, there will be other vectors with the property that N·N = 0. We say that such a vector N is null or equivalently lightlike.
I define the gorm of a vector to be dot the product of a vector with itself. The gorm is bilinear, unlike the norm which is linear. For a spacelike vector, the norm is √(S·S), and corresponds to the notion of proper length. Meanwhile, for a timelike vector, the norm is √(−T·T), which corresponds to the notion of proper time interval. For a null vector, the norm is zero.
For a more general definition of gorm, see section 2.17.
We define the reverse of a clif as follows: Express the clif as a sum of products, and within each term, reverse the order of the factors. For example, the reverse of (P + Q∧R) is (P + R∧Q), where P, Q, and R are vectors (or perhaps scalars).
Notation: For any clif C, the reverse of C is denoted C^{∼}
Reverse has no effect on individual scalars or vectors, but is important for bivectors and highergrade objects. Reverse plays an important role in the description of rotations, as discussed in reference 2. It also appears in the general definition of gorm, as discussed in section 2.17
Note: If you are familiar with complex numbers, you can understand the reverse as a generalization of the notion of complex conjugate. See reference 1 for details.
In all generality, the gorm of any clif C is formed by multiplying C by the reverse of C, and keeping the scalar part of the product. That is:
gorm(C) := ⟨C^{∼} C⟩_{0} (31) 
For example, if C = a + b γ_{1} + c γ_{2} + d γ_{1} γ_{2}, then the gorm of C is a^{2} + b^{2} + c^{2} + d^{2} ... assuming γ_{1} and γ_{2} are spacelike basis vectors, as discussed in section 2.18. You can almost consider this a generalization of the Pythagorean formula, where in a nonabstract sense γ_{1} is perpendicular to γ_{2}, and in some much more abstract sense the terms of each grade are “orthogonal” to the terms of every other grade.
Given any set of d linearlyindependent nonnull vectors, we can create an orthonormal basis set, i.e. a set of d mutuallyorthogonal unit vectors. Actually we can create arbitrarily many such sets.
Proof by construction: Use the GramSchmidt renormalization algorithm. That is, use formulas like equation 10 to project out mutuallyorthogonal components. Then divide by the norm to normalize them.
In Minkowski spacetime, any such basis will have the following properties:
γ_{0} γ_{0} = −1 (32) 
γ_{1} γ_{1} = γ_{2} γ_{2} = γ_{3} γ_{3} = +1 (33) 
and
γ_{i} γ_{j} = − γ_{j} γ_{i} for all i ≠ j (34) 
In this basis set, γ_{0} is the timelike unit vector, while γ_{1}, γ_{2}, and γ_{3} are the spacelike unit vectors.
Remark: The minus sign in equation 32 stands in contrast to the plus sign in equation 33. This one difference in signs is essentially the only thing that sets special relativity apart from ordinary Euclidean geometry. This point is discussed more fully in reference 9.
In ordinary Euclidean space, the story is the same, except there are no timelike vectors, so you simply forget about γ_{0} and equation 32.
Given a basis, we can write any arbitrary vector V as a linear combination of the basis vectors:
V = a γ_{0} + b γ_{1} + c γ_{2} + d γ_{3} (35) 
for suitable scalars a, b, c, and d.
Terminology: These scalars (a, b, c, and d) are sometimes called the components of V in the chosen basis. They can also be called the matrix elements of V in the chosen basis.
Terminology: The vector a γ_{0} is sometimes called the component of V in the chosen γ_{0} direction (and similarly for the other terms on the RHS of equation 35). Such a vector can also be called the projection of V onto the chosen directions.
It is usually obvious from context which definition of “component” is intended. If you want to avoid ambiguity in your writing, you can avoid the word “component” and instead say “matrix element” or “projection” as appropriate.
We can also write the expansion of V as
V = V^{0} γ_{0} + V^{1} γ_{1} + V^{2} γ_{2} + V^{3} γ_{3} (36) 
where again the V^{i} are called the components (or matrix elements) of V in the chosen basis.
Beware: Even though V^{i} is a component of the vector V and is a scalar, please do not think of γ_{i} in the same way. Each γ_{i} is a vector unto itself, not a scalar. The i in γ_{i} tells which vector, whereas the i in V^{i} tells which component of the vector. There is no advantage in imagining some supervector that has the γ_{i} vectors as its components.
Given two vectors P and Q, the dot product can be expressed in terms of their components as follows:
P·Q = −P^{0} Q^{0} + P^{1} Q^{1} + P^{2} Q^{2} + P^{3} Q^{3} (37) 
assuming timelike γ_{0} and spacelike γ_{1}, γ_{2}, and γ_{3}.
Note that this is not the definition of dot product; it is merely a consequence of the earlier definition of dot product (equation 5) and the definition of components (equation 37).
In particular: Nonexperts sometimes think the dot product is “defined” by adding up the product of corresponding components, but that is not true in general, as you can see from the minus sign in front of the first term in equation 37. The bogus definition is reinforced by software library routines that compute a socalled “dot product” by blithely multiplying corresponding elements. (If all the basis vectors are spacelike, then you can get away with just multiplying the corresponding components, but keep in mind that that’s not the definition, and not the general rule.)
Useful tutorials on various aspects of Clifford algebra and its application to physics include reference 10, reference 11, and reference 8.
The number of components required to describe a clif depends on the number of dimensions involved. The first few cases are shown in this table:
 (38) 
where s means scalar, v means vector, b means bivector, t means trivector, and q means quadvector. You can see that it takes the form of Pascal’s triangle. On each row, the total number of components is 2^{D}.
When we speak of “the” dimension, it refers to the dimensionality of whatever space you are actually using. This includes the case where, for whatever reason, attention is restricted to some subspace of the natural universe. For example, if your universe contains three dimensions, but you are only considering a single plane of rotation, then the D=2 description applies. Similarly, if you live in fourdimensional spacetime, but are only considering rotations in the three spacelike directions, then the D=3 description applies.
To say the same thing another way, Clifford algebra is remarkably agnostic about the existence (or nonexistence) of dimensions beyond the ones you are actually using. (This makes the geometric product much more elegant than the oldfashioned vector cross product, which requires you to think about a third dimension, even if you only started out with two dimensions.)
If you find that you need D dimensions to describe the laws of physics, that sets a lower bound on the dimensionality of the universe you live in. It is not an upper bound, i.e. it provides not the slightest evidence against the existence of additional, unseen dimensions, such as arise in string theory.
This section doesn’t explain the physics. Sorry. The purpose here is just to collect some useful formulas. In particular, if you ever forget the sign conventions, this may serve as a reminder.
The angular momentum of a pointlike object is given by
 (39) 
where r is the object’s position and p is its momentum (i.e. its ordinary linear momentum).
The fundamental laws of physics say that both linear momentum and angular momentum are conserved.
The object’s angular velocity is given by
 (40) 
where v is the velocity.
It is often useful to find the linear velocity (v) in terms of the angular velocity (ω) and the position:
 (41) 
as you can verify by plugging the definition of ω (from equation 40) into the RHS of equation 41. You will also need the definition of wedge product, i.e. equation 4. Beware the minus sign in the last line of equation 41; multiplying a vector times a bivector is not commutative.
This section should be skipped on first reading. For most purposes, the dot product – as defined by equation 20 or equation 42d – is the only type of dot product you need to know. On the other hand, there are occasionally situations where one of the other products defined in equation 42 allow something to be expressed more simply; see section 5.1 for an example.
By way of background, recall the definition of the product P := A·B from elementary vector algebra. This is known as the dot product, inner product, or scalar product. The elementary definition applies only to the case where A and B are ordinary grade=1 vectors. In this case, the product has several interesting properties:
We wish to generalize this notion of product, so that it applies to arbitrary clifs. There are several ways of doing this, depending on which properties we are most interested in preserving.

Loosely speaking, these all try to capture the idea that the contraction is the lowestgrade piece of the geometric product. The Hestenes inner product is almost never a good idea; we mention it here just so you don’t get confused if you see it in the literature. Beware that people who like the Hestenes inner product typically write it with a simple •, so it can easily be confused with the more conventional dot product. Similarly, people who like the scalar product typically write it with a simple •, so now we have three different products represented by the same symbol. In cases where it is necessary to distinguish the conventional dot product (as defined in equation 20 or equation 42d) from other things, it can be called the “fat” dot product. For details, see reference 12.
Beware that the symbol ⌞ resembles the letter “L” but it stands for the right (not left) contraction. I don’t know of any good mnemonic for this.
If A and B are not homogeneous, we extend these definitions by using the fact that the product distributes over addition. That gives us the following formulas:

The Hodge dual is a linear operator. In d dimensions, it maps a blade of grade g into a blade of d−g and vice versa. I call this the the gradeflipping property. Two particularlycommon cases are illustrated in figure 5 and figure 6.
As a first example, the Hodge dual always converts a scalar into the corresponding pseudoscalar. As another example, in d=3, it converts a bivector to a certain “corresponding” pseudovector. Meanwhile, in d=4, it converts a vector into the “corresponding” pseudovector.
Let V be any vector (grade=1) and let C be an arbitrary blade (grade=g). Then the wedge product V∧C will have grade=g+1, unless the product vanishes. We can summarize this by saying that wedgemultiplying by a vector raises the grade by 1, roughly speaking.
Similarly, let V be any vector (grade=1) as before, and let D be an arbitrary blade that is not a scalar (grade=g, where g>0). Then the dot product V·D will have grade=g−1, unless the product vanishes. We can summarize this by saying that dotmultiplying by a vector lowers the grade by 1, roughly speaking.
Optional tangential remark: We can simplify the previous paragraph by using the leftcontraction operator defined in section 4. Let V be any vector (grade=1) as before, and let C be an arbitrary blade (grade=g), scalar or otherwise. Then the leftcontraction V ⌟ C will have grade=g−1, unless the product vanishes. We can summarize this by saying that leftcontracting with a vector lowers the grade by 1, roughly speaking.
Combining these ideas, we can say that wedgemultiplying is in some sense the gradeflipped version of dotmultiplying. That is, we should be able to define a correspondence, where wedgemultiplying moves us downward in the left column of figure 6, while dotmultiplying moves us upward in the right column. In fact, we can use this idea to define the Hodge dual. (Note that figure 5 and figure 6 do not fully define the Hodge dual; they merely describe some of its features.)
Start with a blade B with grade=k. Let X denote the Hodge dual of B. We are not yet able to calculate X, but we know that whatever it is, it must have grade d−k. We now find some other blade A with the same grade as the original B, namely grade=k. The wedge product A∧X will be a pseudoscalar. That is to say, it will have the largest possible grade, namely grade=d.
Meanwhile, we can also form the dot product, A·B. This will have the smallest possible grade, namely grade=0.
Finally, we choose some unit pseudoscalar, which we denote i. If we have an ordered set of basis vectors, the obvious choice is to multiply all of them together in order, so that i = γ_{1}γ_{2}⋯γ_{d}. (Note that by assigning an order to the basis vectors, we introduce a notion of chirality. This is significant addition, insofar as the foundations of Clifford algebra do not require chirality, as discussed in section 2.13.)
At this point we have all the tools needed to define the Hodge dual.

where B§_{i} denotes the Hodge dual of B, namely the Hodge dual with respect to i. Under mild conditions, there is always one and only one X that satisfies equation 44a, so the Hodge dual exists and is unique. We require that in some basis, every basis vector that appears in B must also appear in i.
In a fairly wide range of practical applications, there is an obvious choice for i, which is then called the “preferred” pseudoscalar. In is therefore conventional to simplify the notation by writing the dual of B as *B, where this “*” is a unary prefix operator. This is conventional, but it is not entirely wise. That’s because sometimes we want to think about physics in fourdimensional Minkowski spacetime, and sometimes we want restrict attention to threedimensional Euclidean space. The Hodge dual is very different in these two cases, because the relevant pseudoscalar is different.
Example: Suppose B=7 (i.e. a scalar). Then B§_{i}=7i (i.e. a pseudoscalar). This is true in any number of dimensions, from d=1 on up.
Example: In d=3, suppose B = 5γ_{1} γ_{2} (i.e. a bivector in the xy plane). Then B§_{i} = 5γ_{3} = [0, 0, 5] (i.e. a vector in the z direction). This assumes i = γ_{1}γ_{2}γ_{3}.
Remark: equation 44 provides an implicit definition. There exist explicit cutanddried algorithms for calculating the Hodge dual of B, especially if B is known in terms of components in some basis. See the discussion in reference 13, or see the actual code in reference 14.
If you’re not using the Hodge dual, Clifford algebra has the following remarkable property, which I find starkly beautiful: Suppose you are working with three basis vectors, γ_{1}, γ_{2}, and γa. You know that your space must have at least three dimensions, but you have no way of knowing whether that is merely a subspace of some much larger space. That is, there could be other basis vectors that you do not know about, and do not need to know about. That’s because Clifford algebra is closed under the usual operations (dot product, wedge product, addition, et cetera). Furthermore, you do not need to arrange your basis vectors in order. You do not need to know whether γ_{a} comes before the others, or after, or in between. You do not have a “righthand rule” and you do not need one.
I call this property “subspace freedom”.
In contrast, if you write the Hodge dual in the form *B, it does not permit subspace freedom. That’s because *B implicitly depends on some specific, chosen, “preferred” unit pseudovector.
According to the usual way of thinking about *B, the grade of the pseudoscalar tells you exactly how many basis vectors there are. Furthermore, the choice of pseudoscalar defines a notion of righthanded versus lefthanded, because if you perform an odd permutation of the basis vectors, the sign of the “preferred” pseudoscalar changes.
In contrast, writing the Hodge dual in the form B§_{i} reveals the dependence on i and preserves subspace freedom. There could be other basis vectors you don’t know about and don’t need to know about, so long as they do not appear in B or in i.
In this subsection, we restrict attention to d=3. We choose our preferred unit pseudoscalar in the obvious way, namely i=γ_{1}γ_{2}γ_{3}.
The cross product A×B is a pseudovector. It can be calculated in terms of the bivector A∧B as follows:

where in equation 45a the operator §_{i} is our representation for the Hodge dual, as defined in section 5.1. Meanwhile, equation 45b expresses exactly the same thing, using the more conventional unary prefix * notation.
The Hodge dual operator is a linear operator. That means that if you are not planning on doing anything nonlinear with your cross products, you can pretty much ignore the Hodge dual operator in equation 45. That leaves us with the following quickanddirty recipe:
 (46) 
As part of this recipe, replace the notion of “axis of rotation” with “plane of rotation”. For example, rather than rotation around the Z axis, think in terms of rotation in the XY plane.
You need to stick with the more general expression in equation 45 if you plan to multiply your bivectors by vectors or highergrade clifs. As an example, consider the triple scalar product, representing the volume of the parallelepiped spanned by three vectors. We can write that as:
 (47) 
In this case, A×B·C is strictly equal to A∧B∧C. They are two ways of calculating the same pseudoscalar. For more about the volume of a parallelepiped, see reference 7.
In more advanced situations such as electrodynamics, you often find that there is an equation involving a cross product and a “similar” equation involving a dot product. In this case you may need to combine the two equations and replace both products with the full geometric product. This generally requires more than a recipe, i.e. it may require actually understanding what the equations mean. For electrodynamics in particular, the right answer involves promoting the equations from 3D to 4D, and replacing two vector fields by one bivector field. See reference 4.
I prefer the wedge product for several reasons
First and foremost is the simplest of practical pedagogical reasons: I can get good results using wedge products. The more elementary the context, and the more unprepared the students, the more helpful wedge products are. I can visualize the wedge product, and I can get the students to visualize it. (This stands in stark contrast to the cross product, which tends to be very mysterious to students.)
Specifically: Consider gyroscopic precession. I have done the pedagogical experiment more times than I care to count. I have tried it both ways, using cross products (pseudovectors) and/or using wedge products (bivectors). Precession can be understood using little more than the addition of bivectors, adding them edgetoedge as described in section 2.3. I can use my hands to represent the bivectors to be added, or (even better) I can show up with simple cardboard props.
Getting students to visualize angular momentum as a bivector in the plane of rotation takes no time at all. In contrast, getting them to visualize it as a pseudovector along the axis of rotation is a big production; even a bright student is going to struggle with this, and the notsobright students are never going to get it.
Maybe this just means I’m doing a lousy job of explaining cross products, but even if that’s true, I’ll bet there are plenty of teachers out there who find themselves in the same situation and would benefit from taking the bivector approach.
Another reason for preferring the geometrical approach, i.e. the Clifford algebra approach, is that (compared to cross products) it does a much better job of modeling the symmetries of the real world.
Remember, the real world is what it is and does what it does. When we write down an equation, it may or may not be an apt model of the real world. Often an equation involving a cross product will have a chirality (“handedness”) to it, even when the realworld physics is not chiral. For more on this, see reference 3.
Additional reasons for preferring the geometric approach have to do with its connections to other mathematical and physical ideas, as discussed in section 1. (This is in contrast to the cross product, which is not nearly so extensible.)
There are always multiple ways of presenting the same material. In the case of Clifford algebra, it would have been possible to leap to the idea of a basis set very early on. The sequence would have been: (a) establish a few fundamental notions; (b) set forth the behavior of the basis vectors according to equation 33 and equation 34; (c) express all vectors, bivectors, etc. in terms of their components relative to this basis; and (d) derive the main results in terms of components. Let’s call this the basis+components approach.
Some students prefer the basis+components approach because it is what they are expecting. They think a vector, by definition, is nothing more than a list of components.
The alternative is to think of a vector as a thing unto itself, as an object with geometric properties, independent of any basis. Let’s call this the geometric approach.
We must ask the question, which approach is more elementary, and which approach is more sophisticated? Also, which approach is more abstract, and which approach is more closely tied to physical reality?
Actually those are trick questions; they look like dichotomies, but they really aren’t. The answer is that the geometric approach is more physical and more abstract. It is more elementary and more sophisticated.
You can use a pointed stick as a physical model of a vector, independent of any basis. Wave it around in front of the class.  Show how vectors can be added by putting them tiptotail. 
Use a flat piece of cardboard as a physical model of a bivector.  Show how bivectors can be added by putting them edgetoedge. 
In physics, almost everything worth knowing can be expressed independently of any basis. You should be suspicious of anything that appears to depend on a particular chosen basis. For more about this, see reference 15. Observe that everything in section 2 is done without reference to any basis; the definition of “basis” and “components” are not even mentioned until the very end.
The real practical advantage of the basis+components approach is that it is well suited for numerical calculations, including computer programs. Reference 2 includes a program that is useful for keeping track of compound rotations in D=3 space.
Consider the contrast:
The geometric approach is physical yet algebraic, axiomatic, and abstract; it is elementary yet sophisticated.  The basis+components approach is more numerical, i.e. more suited for computer programs. 
Some students have prior familiarity with one approach, or the other, or both, or neither.
By longstanding convention, the term “outer product” of two vectors (V and W) refers to the tensor product, namely
 (48) 
where T is a secondrank tensor.
If we expand things in terms of components, relative to some basis, we can write:
 (49) 
For example:
 (50) 
In general, T itself has no particular symmetry, but we can pick it apart into a symmetric piece plus an antisymmetric piece:
 (51) 
The second term in on the RHS of equation 51 is the wedge product V∧W. The Trace of the first term is the dot product.
Beware: Reference 8 introduces the term “antisymmetric outer product” to refer to the wedge product, and then shorthands it as “outer product”, which is highly ambiguous and nonstandard. It conflicts with the longstanding conventional definition of outer product, which consists of all of T, not just the antisymmetric part of T.
I wrote a “Clifford algebra desk calculator” program. It knows how to do addition, subtraction, dot product, wedge product, full geometric product, reverse, hodge dual, and so forth. Most of the features work in arbitrarily many dimensions.
Here is the program’s help message. See also reference 14.
Desk calculator for Clifford algebra in arbitrarily many Euclidean dimensions. (No Minkowski space yet; sorry.) Usage: ./cliffer [options] Commandline options include h print this message (and exit immediately). v increase verbosity. i fn take input from file 'fn'. pre fn take preliminary input from file 'fn'.  take input from STDIN If no input files are specified with i or , the default is an implicit ''. Note that i and pre can be used multiple times. All pre files are processed before any i files. Advanced usage: If you want to make an input file into a selfexecuting script, you can use "#! /path/to/cliffer i" as the first line. Similarly, if you want to do some initialization and then read from standard input, you can use "#! /path/to/cliffer pre" as the first line. Ordinary usage example: # compound rotation: two 90 degree rotations # makes a 120 degree rotation about the 1,1,1 diagonal: echo e "1 0 0 90° vrml 0 0 1 90° vrml mul @v"  cliffer Result: 0.57735 0.57735 0.57735 2.09440 = 120.0000° Explanation: *) Push a rotation operator onto the stack, by giving four numbers in VRML format X Y Z theta followed by the "vrml" keyword. *) Push another rotation operator onto the stack, in the same way. *) Multiply them together using the "mul" keyword. *) Pop the result and print it in VRML format using the "@v" keyword On input, we expect all angles to be in radians. You can convert from degrees to radians using the "deg" operator, which can be abbreviated to "°" (the degree symbol). Hint: Alt0 on some keyboards. As a special case, on input, a number with suffix "d" (with no spaces between the number and the "d") is converted from degrees to radians. echo "90° sin @"  cliffer echo "90 ° sin @"  cliffer echo "90d sin @"  cliffer are each equivalent to echo "pi 2 div sin @"  cliffer Input words can be spread across as many lines (or as few) as you wish. If input is from an interactive terminal, any error causes the rest of the current line to be thrown away, but the program does not exit. In the noninteractive case, any error causes the program to exit. On input, a comma or tab is equivalent to a space. Multiple spaces are equivalent to a single space. Note on VRML format: X Y Z theta [X Y Z] is a vector specifying the axis of rotation, and theta specifies the amount of rotation around that axis. VRML requires [X Y Z] to be normalized as a unit vector, but we are more tolerant; we will normalize it for you. VRML requires theta to be measured in radians. Also note that on input, the VRML operator accepts either four numbers, or one 3component vector plus one scalar, as in the following example. Same as previous example, with more output: echo e "[1 0 0] 90° vrml dup @v dup @m [0 0 1] 90° vrml rev mul dup @v @m"  cliffer Result: 1.00000 0.00000 0.00000 1.57080 = 90.0000° [ 1.00000 0.00000 0.00000 ] [ 0.00000 0.00000 1.00000 ] [ 0.00000 1.00000 0.00000 ] 0.57735 0.57735 0.57735 2.09440 = 120.0000° [ 0.00000 0.00000 1.00000 ] [ 1.00000 0.00000 0.00000 ] [ 0.00000 1.00000 0.00000 ] Even fancier: Multiply two vectors to create a bivector, then use that to crank a vector: echo e "[ 1 0 0 ] [ 1 1 0 ] mul normalize [ 0 1 0 ] crank @" \  ./cliffer Result: [1, 0, 0] Another example: Calculate the angle between two vectors: echo e "[ 1 0 0 ] [ 1 1 0 ] mul normalize rangle @a"  ./cliffer Result: 2.35619 = 135.0000° Example: Powers: Exponentiate a quaternion. Find rotor that rotates only half as much: echo e "[ 1 0 0 ] [ 0 1 0 ] mul 2 mul dup rangle @a " \ " .5 pow dup rangle @a @"  ./cliffer Result: 1.57080 = 90.0000° 0.78540 = 45.0000° 1 + [0, 0, 1]§ Example: Take the 4th root using pow, then take the fourth power using direct multiplication of quaternions: echo "[ 1 0 0 ] [ 0 1 0 ] mul dup @v .25 pow dup @v dup mul dup mul @v"  ./cliffer Result 0.00000 0.00000 1.00000 3.14159 = 180.0000° 0.00000 0.00000 1.00000 0.78540 = 45.0000° 0.00000 0.00000 1.00000 3.14159 = 180.0000° More systematic testing: ./cliffer.test1 The following operators have been implemented: help help message listops list all operators === Unary operators pop remove top item from stack neg negate: multiply by 1 deg convert number from radians to degrees dup duplicate top item on stack gorm gorm i.e. scalar part of V~ V norm norm i.e. sqrt(gorm} normalize divide top item by its norm rev clifford '~' operator, reverse basis vectors hodge hodge dual aka unary '§' operator; alt' on some keyboards gradesel given C and s, find the grades part of C rangle calculate rotor angle === Binary operators exch exchange top two items on stack codot multiply corresponding components, then sum add add top two items on stack sub sub top two items on stack mul multiply top two items on stack (in subspace if possible) cmul promote A and B to clifs, then multiply them div divide clif A by scalar B dot promote A and B to clifs, then take dot product wedge promote A and B to clifs, then take wedge product cross the hodge of the wedge (familiar as cross product in 3D) crank calculate R~ V R pow calculate Nth power of scalar or quat sqrt calculate square root of power of scalar or quat === Constructors [ mark the beginning of a vector ] construct vector by popping to mark unpack unpack a vector, quat, or clif; push its contents (normal order) dimset project object onto Ndimensional Clifford algebra unbave top unit basis vector in N dimensions ups unit pseudoscalar in N dimensions pi push pi onto the stack vrml construct a quaternion from VRML representation x,y,z,theta clif take a vector in D=2**n, construct a clif in D=n Note: You can do the opposite via '[ exch unpack ]' === Printout operators setbasis set basis mode, 0=abcdef 1=xyzabc dump show everything on stack, leave it unchanged @ compactly show item of any type, D=3 (then remove it) @m show quaternion, formatted as a rotation matrix (then remove it) @v show quaternion, formatted in VRML style (then remove it) @a show angle, formatted in radian and degrees (then remove it) @x print clif of any grade, row by row === Math library functions: sin cos tan sec csc cot sinh cosh tanh asin acos atan asinh acosh atanh ln log2 log10 exp atan2 