We saw in the previous section on dot
products that the dot
product takes two vectors and produces a scalar, making it an example
of a scalar product. In this section, we will introduce a vector
product, a multiplication rule that takes two vectors and produces a new
vector. We will find that this new operation, the cross product, is
only valid for our 3-dimensional vectors, and cannot be defined in the 2-
dimensional case. The reasons for this will become clear when we discuss the
kinds of properties we wish the cross product to have.

Rotational Invariance

One important feature of the dot product which we didn't mention in the previous
section is its invariance under rotations. In other words, if we take a
pair of vectors in the plane and rotate them both by the same angle (imagine,
for instance, that the vectors are sitting on a record, and rotate the record),
their dot product will remain the same. Consider the length of a single vector
(which is given by the dot product): if the vector gets rotated about the origin
by some angle, its length will not change--even though its direction can change
quite dramatically! Similarly, from the geometric formula for the dot product,
we see that the result depends only on the lengths of the two vectors and the
angle between them. None of these quantities changes when we rotate the two
vectors together, so neither can their dot product. This is what we mean when
we say that the dot product is invariant under rotations.

Rotational invariance ends up being a very important property in physics.
Imagine writing down vector equations to describe some physical situation taking
place on a table. Now rotate the table (or keep the table fixed, and rotate
yourself by some angle around the table). You haven't really changed anything
about the physics on the table by simply turning everything by some fixed angle.
Because of this, you should expect your equations to retain their form. This
means that if these equations involve products of vectors, these products better
be rotationally invariant. The dot product has already passed this test, as we
noted above. We now want to require the same of the cross product.

Making the requirement of rotational invariance more stringent for the cross
product, we need the cross product of two vectors to yield another
vector. Consider, for instance, two 3-dimensional vectors u and v in
a plane (two non-parallel vectors always define a plane, in the same way that
two lines do. If we rotate this plane, the vectors will change direction, but
we don't want the cross product w = u×v to change at all. However, if w has
any non-zero components in the plane of u and v, those components will
necessarily change under rotation (they get rotated just like everything else).
The only vectors that won't change at all under a rotation of the u-v plane
are those vectors that are perpendicular to the plane. Hence, the
cross product of two vectors u and v must give a new vector which is
perpendicular to both u and v.

This simple observation actually goes a long way towards constraining our
options for how we can define the cross product. For instance, we can see
immediately that it is not possible to define a cross product for two-
dimensional vectors, since there is no direction perpendicular to the plane
of two-dimensional the vectors! (We'd need a third dimension for that).

Now that we know the direction in which the cross product of two vectors
points, the magnitude of the resulting vector remains to be specified.
If I take the cross product of two vectors in the x-y plane, I now know that
the resulting vector should point purely in the z-direction. But should it
point upwards (i.e. lie along the positive z-axis) or should it point downwards? How
long should it be?

Let's begin by defining the cross product for the unit vectors i, j, and k. Since all
vectors can be decomposed in terms of unit vectors (see Unit vectors), once
we've defined the cross products for this special case it will be easy to extend the definition to include all vectors. As we
noted above, the cross product between i and j (since they both lie in the x-y plane) must point
purely in the z-direction. Hence:

i×j = ck

for some constant c. Because later on we will want the magnitude of the
resultant vector to have geometric significance, we need ck to have unit length. In other words, c can be
either +1 or -1. Now we make a completely arbitrary choice in order to accord with convention: we choose c = + 1. The fact
that we have chosen c to be positive is known as The Right-Hand Rule (we could just as easily have chosen c = - 1, and
the math would all work out to be the same as long as we were consistent--but we do have to choose one or the other,
and there's no use going against what everyone else does.) It turns out that in order to be consistent with the Right-Hand
Rule, all of the cross products between unit vectors are uniquely determined:

i×j

=

k = - j×i

j×k

=

i = - k×j

k×i

=

j = - i×k

In particular, notice that the order of the vectors within the cross products
holds significance. In general, u×v = - v×u. From here we can see that the cross product
of a vector with itself is always zero, since by the above rule u×u = - u×u, which means that
both sides must vanish for equality to hold. We can now complete our list of cross products between
unit vectors by observing that:

i×i = j×j = k×k = 0

To take the cross product of two general vectors, we first decompose the vectors
using the unit vectors i, j, and k, and then proceed
to distribute the cross product across the sums, using the above rules to do the cross
products between unit vectors. We can do this for arbitrary vectors u = (u_{1}, u_{2}, u_{3}) and v = (v_{1}, v_{2}, v_{3}) to get a general formula:

Unfortunately, this is as easy as it gets when it comes to writing out the cross
product explicitly in terms of vector components. It is probably a good to keep
this formula handy until you become used to computing vector cross products.

Geometric Formula for Cross Product

Fortunately, as is the case with the dot product, there is a simple geometric
formula for computing the cross product of two vectors, if their respective
lengths and the angle between them is known. Consider the cross product of two
(not necessarily unit-length) vectors that lie purely along the x and y axes (as
i and j do). We can thus write the vectors as u = ai and
v = bj, for some constants a and b. The cross product u×v is thus
equal to

u×v = ab(i×j) = abk

Notice that the magnitude of the resultant vector is the same as the area of the
rectangle with sides u and v! As promised above, the magnitude of the cross
product between two vectors, | u×v|, has a geometric interpretation. In
general it is equal to the area of the parallelogram having the two given
vectors as its sides (see ).

From basic geometry, we know that this area is given by
area= | u|| v| sinθ, where | u| and | v| are the lengths of the sides of
the parallelogram, and θ is the angle between the two vectors. Notice
that when the two vectors are perpendicular to each other, θ =90 degrees,
so sinθ =1 and we recover the familiar formula for the area of a square.
On the other hand, when the two vectors are parallel, θ =0 degrees, and
sinθ=0, meaning the area vanishes (as we expect). In general, then, we
find that the magnitude of the cross product between two vectors u and v
that are separated by an angle θ (going clockwise from u to v, as
specified by the Right-Hand Rule) is given by:

| u×v| = | u|| v| sinθ

In particular, this means that for two parallel vectors the cross product equals
0.

Cross Product Summary

In summary, the cross product of two vectors is given by: