2.4 The cross product  (Page 2/16)

Notice what this means for the direction of $\text{v}\times \text{u}.$ If we apply the right-hand rule to $\text{v}\times \text{u},$ we start with our fingers pointed in the direction of $\text{v},$ then curl our fingers toward the vector $\text{u}.$ In this case, the thumb points in the opposite direction of $\text{u}\times \text{v}.$ (Try it!)

Anticommutativity of the cross product

Let $\text{u}=⟨0,2,1⟩$ and $\text{v}=⟨3,\mathrm{-1},0⟩.$ Calculate $\text{u}\times \text{v}$ and $\text{v}\times \text{u}$ and graph them.

We have

$\begin{array}{ccc}\hfill \text{u}\times \text{v}& =\hfill & ⟨\left(0+1\right),\text{−}\left(0-3\right),\left(0-6\right)⟩=⟨1,3,\mathrm{-6}⟩\hfill \\ \hfill \text{v}\times \text{u}& =\hfill & ⟨\left(\mathrm{-1}-0\right),\text{−}\left(3-0\right),\left(6-0\right)⟩=⟨\mathrm{-1},\mathrm{-3},6⟩.\hfill \end{array}$

We see that, in this case, $\text{u}\times \text{v}=\text{−}\left(\text{v}\times \text{u}\right)$ ( [link] ). We prove this in general later in this section.

Got questions? Get instant answers now!

Got questions? Get instant answers now!

The cross products of the standard unit vectors $\text{i},\text{j},$ and $\text{k}$ can be useful for simplifying some calculations, so let’s consider these cross products. A straightforward application of the definition shows that

(The cross product of two vectors is a vector, so each of these products results in the zero vector, not the scalar $0.)$ It’s up to you to verify the calculations on your own.

Furthermore, because the cross product of two vectors is orthogonal to each of these vectors, we know that the cross product of $\text{i}$ and $\text{j}$ is parallel to $\text{k}.$ Similarly, the vector product of $\text{i}$ and $\text{k}$ is parallel to $\text{j},$ and the vector product of $\text{j}$ and $\text{k}$ is parallel to $\text{i}.$ We can use the right-hand rule to determine the direction of each product. Then we have

These formulas come in handy later.

As we have seen, the dot product is often called the scalar product because it results in a scalar. The cross product results in a vector, so it is sometimes called the vector product    . These operations are both versions of vector multiplication, but they have very different properties and applications. Let’s explore some properties of the cross product. We prove only a few of them. Proofs of the other properties are left as exercises.

Proof

For property $\text{i}.,$ we want to show $\text{u}\times \text{v}=\text{−}\left(\text{v}\times \text{u}\right).$ We have

Unlike most operations we’ve seen, the cross product is not commutative. This makes sense if we think about the right-hand rule.