General Mechanics/Cross Product

There are two ways to multiply two vectors together, the dot product and the cross product. We have already studied the dot product of two vectors, which results in a scalar or single number.

The cross product of two vectors results in a third vector, and is written symbolically as follows:

${\vec {A}}\times {\vec {B}}$

The cross product of two vectors is defined to be perpendicular to the plane defined by these vectors. However, this doesn't tell us whether the resulting vector points upward out of the plane or downward. This ambiguity is resolved using the right-hand rule:

1. Point the uncurled fingers of your right hand along the direction of the first vector ${\vec {A}}$ .
2. Rotate your arm until you can curl your fingers in the direction of the second vector ${\vec {B}}$ .
3. Your stretched out thumb now points in the direction of the cross product vector ${\vec {A}}\times {\vec {B}}$ .

The magnitude of the cross product is given by

$|{\vec {A}}\times {\vec {B}}|=|{\vec {A}}||{\vec {B}}|\cdot \sin(\theta )$

where $|{\vec {A}}|$ and $|{\vec {B}}|$ are the magnitudes of ${\vec {A}},{\vec {B}}$ , and $\theta$ is the angle between these two vectors. Note that the magnitude of the cross product is zero when the vectors are parallel or anti-parallel, and maximum when they are perpendicular. This contrasts with the dot product, which is maximum for parallel vectors and zero for perpendicular vectors.

Notice that the cross product does not commute, i.e. the order of the vectors is important. In particular, it is easy to show using the right-hand rule that

${\vec {A}}\times {\vec {B}}=-{\vec {B}}\times {\vec {A}}$

An alternate way to compute the cross product is most useful when the two vectors are expressed in terms of components,

${\vec {C}}={\vec {A}}\times {\vec {B}}={\begin{vmatrix}\mathbf {i} &\mathbf {j} &\mathbf {k} \\A_{x}&A_{y}&A_{z}\\B_{x}&B_{y}&B_{z}\end{vmatrix}}$

where the determinant is expanded as if all the components were numbers, giving

${\begin{aligned}C_{x}&=A_{y}B_{z}-A_{z}B_{y}\\C_{y}&=A_{z}B_{x}-A_{x}B_{z}\\C_{z}&=A_{x}B_{y}-A_{y}B_{x}\end{aligned}}$

Note how the positive terms possess a forward alphabetical direction, xyzxyzx... (with x following z):

With the cross product we can also multiply three vectors together, in two different ways.

We can take the dot product of a vector with a cross product, a triple scalar product,

${\vec {A}}\cdot ({\vec {B}}\times {\vec {C}})=({\vec {A}}\times {\vec {B}})\cdot {\vec {C}}$

The absolute value of this product is the volume of the parallelpiped defined by the three vectors ${\vec {A}},{\vec {B}},{\vec {C}}$

Alternately, we can take the cross product of a vector with a cross product, a triple vector product, which can be simplified to a combination of dot products.

${\vec {A}}\times ({\vec {B}}\times {\vec {C}})=({\vec {A}}\cdot {\vec {C}}){\vec {B}}-({\vec {A}}\cdot {\vec {B}}){\vec {C}}$

This form is easier to do calculations with.

The triple vector product is not associative.

${\vec {A}}\times ({\vec {B}}\times {\vec {C}})\neq ({\vec {A}}\times {\vec {B}})\times {\vec {C}}$

A nice as well as a useful way to denote the cross product is using the indicial notation

${\vec {C}}={\vec {A}}\times {\vec {B}}=\epsilon ^{ijk}\,A_{j}\,B_{k}\,{\hat {e}}_{i}$

where $\epsilon ^{ijk}$ is the Levi-Civita alternating symbol and ${\hat {e}}_{i}$ is either of the unit vectors ${\vec {\imath }},{\vec {\jmath }},{\vec {k}}$ . (A good exercise to convince yourself would be to use this expression and see if you can get ${\vec {C}}={\vec {A}}\times {\vec {B}}$ as defined before.)