Question

What is the cross product of `(- 5 i + 4 j - 5 k)` and `(4 i + 4 j + 2 k)`?

Answer 1

If we call the first vector `vec a` and the second `vec b`, the cross product, `vec a xx vec b` is `(28veci-10vecj-36veck)`.

Explanation:

Sal Khan of Khan academy does a nice job of calculating a cross product in this video: https://www.khanacademy.org/math/linear-algebra/vectors_and_spaces/dot_cross_products/v/linear-algebra-cross-product-introduction

It's something that's easier to do visually, but I'll try to do it justice here:

`vec a = (-5veci+4vecj-5veck)`
`vec b = (4veci+4vecj+2veck)`

We can refer to the coefficient of `i` in `vec a` as `a_i`, the coefficient of `j` in `vec b` as `b_j` and so on.

`vec a xx vec b = (-5veci+4vecj-5veck) xx (4veci+4vecj+2veck)`

Sal's video above and the Wikipedia article on the cross product will do a better job of explaining why the next step is as follows than I can here:

`vec a xx vec b = (a_jb_k-a_kb_j)vec i + (a_kb_i-a_ib_k)vec j+(a_ib_j-a_jb_i)vec k`

`= (4*2-(-5)*4)vec i + ((-5)*4-(-5)*2)vec j+((-5)*4-4*4)vec k = 28vec i -10 vec j -36vec k`