Cross product examples

Example 1

Calculate the cross product between a = (3, - 3, 1) and b = (4, 9, 2).

Solution: The cross product is

$$\left\vert\vphantom{ \begin{array}{ccc} \mathbf{i} & \mathbf{j} & \mathbf{k} 3 & -3 & 1 4 & 9 & 2 \end{array} }\right. \begin{array}{ccc} \mathbf{i} & \mathbf{j} & \mathbf{k} 3 & -3 & 1 4 & 9 & 2 \end{array} \left.\vphantom{ \begin{array}{ccc} \mathbf{i} & \mathbf{j} & \mathbf{k} 3 & -3 & 1 4 & 9 & 2 \end{array} }\right\vert$$ a × b = i(- 3 · 2 - 1 · 9) - j(3 · 2 - 1 · 4) + k(3 · 9 + 3 · 4) = - 15i -2j +39k

Example 2

Calculate the area of the parallelogram spanned by the vectors a = (3, - 3, 1) and b = (4, 9, 2).

Solution: The area is ||a × b||. Using the above expression for the cross product, we find that the area is $\sqrt{{15^2+2^2+39^2}}$ = 5$\sqrt{{70}}$.

Example 3

Calculate the area of the parallelogram spanned by the vectors a = (3, - 3, 1) and c = (- 12, 12, - 4).

Solution:

$$\left\vert\vphantom{ \begin{array}{ccc} \mathbf{i} & \mathbf{j} & \mathbf{k} 3 & -3 & 1 -12 & 12 & -4 \end{array} }\right. \begin{array}{ccc} \mathbf{i} & \mathbf{j} & \mathbf{k} 3 & -3 & 1 -12 & 12 & -4 \end{array} \left.\vphantom{ \begin{array}{ccc} \mathbf{i} & \mathbf{j} & \mathbf{k} 3 & -3 & 1 -12 & 12 & -4 \end{array} }\right\vert$$ a × c = (0, 0, 0)

The magnitude of the zero vector is zero, so the area of the parallelogram is zero. What happened?