Vector product

A vector product is a simple operation between two vectors that returns a new vector that is perpendicular to the two vectors.

What is a vector product

A vector product is defined between two vectors and only in space. The result of a vector product, unlike a scalar product, is again a vector. The result of the vector product of the vectors $\vec{\mathbf{u}}$ and $\vec{\mathbf{v}}$ is the vector $\vec{\mathbf{w}}$, which has the following properties:

$$|\vec{\mathbf{w}}|=|\vec{\mathbf{u}}|\cdot|\vec{\mathbf{v}}|\cdot\sin\alpha,$$

where α is the angle between the vectors $\vec{\mathbf{u}}$ and $\vec{\mathbf{v}}$. Furthermore, the vector $\vec{\mathbf{w}}$ is perpendicular to both vectors $\vec{\mathbf{u}}$ and $\vec{\mathbf{v}}$. The resulting direction then follows the right-hand rule. To distinguish the vector product from the scalar product, we use the sign × for the vector: $\vec{\mathbf{u}} \times \vec{\mathbf{v}}$.

The question shows a possible vector product $\vec{\mathbf{u}} \times \vec{\mathbf{v}}$. The result is the vector $\vec{\mathbf{w}}$, which is perpendicular to both vectors. We are moving in space, so the vectors $\vec{\mathbf{u}}$ and $\vec{\mathbf{v}}$ are in the same plane, and the vector $\vec{\mathbf{w}}$ is perpendicular to this plane.

If one vector is a linear combination of the other, then their vector product is equal to the zero vector. This captures the case where both vectors lie on the same line. If at least one vector is zero, then the resulting product is also zero.

How to find the coordinates of the resultant vector

For now, we can calculate the magnitude of the resultant vector and the direction. However, it is more practical to know the coordinates of such a vector directly. The formula itself looks like this:

$$u\times v=(u_2v_3-v_2u_3, u_3v_1-v_3u_1, u_1v_2-v_1u_2)$$

This formula is quite hard to remember, so there is a tool to help you remember it:

With this table we can already multiply vectors nicely. Example: consider the vectors $\vec{\mathbf{u}}=(4,0,0)$ and $\vec{\mathbf{v}}=(0,5,0)$. These are vectors that lie entirely on the axis x and y, respectively. Their vector product will be equal to:

$$\begin{matrix} \vec{\mathbf{u}}&0&&0&&4&&0\\\hline \vec{\mathbf{v}}&5&&0&&0&&5\\ \vec{\mathbf{u}}\times \vec{\mathbf{v}}&&0&&0&&20 \end{matrix}$$

The resulting vector $\vec{\mathbf{w}}$ has coordinates (0, 0, 20). We can verify that the result is correct by the first definition. It says that

$$|w|=|u|\cdot|v|\cdot\sin\alpha,$$

Our vector $\vec{\mathbf{w}}$ has a size of 20. Thus, according to the formula:

$$20=|u|\cdot|v|\cdot\sin\alpha$$

We add:

$$20=4\cdot5\cdot\sin90^\circ=20\cdot1=20.$$

What's it good for?

Obviously we can use vector product to find a vector that is perpendicular to two other vectors. Again, we can check that this is indeed the case by using the scalar product. The result must be zero. Verification for the previous example:

$$\begin{eqnarray} \vec{\mathbf{u}}\cdot \vec{\mathbf{w}}&=&4\cdot0+0+0\cdot20=0\\ \vec{\mathbf{v}}\cdot \vec{\mathbf{w}}&=&0+5\cdot0+0\cdot20=0 \end{eqnarray}$$

Another example of use is to calculate the area of a parallelogram in space. If we have a parallelogram ABCD in space and vectors $\vec{\mathbf{u}}$ and $\vec{\mathbf{v}}$, which are formed by the sides AB and AD, then the area of the parallelogram is equal to $S=|\vec{\mathbf{u}} \times \vec{\mathbf{v}}|$. If we have a triangle ABC in space, then its content is equal to $S=\frac12|\vec{\mathbf{u}} \times \vec{\mathbf{v}}|$.

Example: calculate the area of the triangle in space that is formed by the vertices: A[1,0,3], B[5, 6, 8] and C[3, 5, 4].

First we need to determine the vectors $\vec{\mathbf{u}}$ and $\vec{\mathbf{v}}$.

$$\begin{eqnarray} u&=&B-A=(4, 6, 5)\\ v&=&C-A=(2, 5, 1) \end{eqnarray}$$

Now we calculate their product:

$$\vec{\mathbf{w}}=\vec{\mathbf{u}}\times \vec{\mathbf{v}}=(-19, 6, 8)$$

And then the magnitude:

$$|\vec{\mathbf{w}}|=\sqrt{(-19)^2+6^2+8^2}=\sqrt{461}\approx21{,}47$$

And finally, we divide the result by two:

$$S(\triangle ABC)=\frac{21{,}47}{2}=10{,}735$$