Question

What is the cross product of `[3,2, 5]` and `[2, -5, 8]`?

Answer 1

By hand and then checked with MATLAB: [41 -14 -19]

Explanation:

When you take a cross product, I feel like it makes things easier to add in the unit vector directions `[ hat i hat j hat k ]` which are in the x, y, and z directions respectively.

We'll use all three since these are 3-D vectors that we're dealing with. If it was 2d you'd only have to use `hati` and `hatj`

Now we set up a 3x3 matrix as follows (Socratic doesn't give me a good way to do multidimensional matrices, sorry!):

`|hati hatj hatk|`
`|3 2 5|`
`|2 -5 8|`

Now, starting at each unit vector, go diagonal from left to right, taking the product of those numbers:

`(2*8)hati (5*2)hatj (3*-5)hatk`

`=16hati 10hatj -15hatk`

Next, take the products of the values going from right to left; again, starting at the unit vector:

`(5*-5)hati (3*8)hatj (2*2)hatk`

`=-25hati 24hatj 4hatk`

Finally, take the first set and subtract the second set from it

`[16hati 10hatj -15hatk]-[-25hati 24hatj 4hatk]`
`=(16-(-25))hati (10-24)hatj (-15-4)hatk`
`=41hati -14hatj -19hatk`

this can now be re-written in matrix form, with `hati`, `hatj`, and `hatk` removed since it's staying a 3-D vector:

`color(red)("[41 -14 -19]")`