Cross product examples
Example 1
Calculate the cross product between a = (3,-3, 1) and b = (4, 9, 2).
Solution: The cross product is
| 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 
= 5
.
Example 3
Calculate the area of the parallelogram spanned by the vectors a = (3,-3, 1) and c = (-12, 12,-4).
Solution:
| a × c | =  | ||
| = (0, 0, 0) |
The magnitude of the zero vector is zero, so the area of the parallelogram is zero. What happened?
