Question

What is the cross product of `[1, -2, -3]` and `[2, -5, 8]`?

Answer 1

The answer is `=〈-31,-14,-1〉`

Explanation:

The cross product of 2 vectors

`veca=〈a_1,a_2,a_3〉`

and `vecb=〈b_1,b_2b_3〉`

is given by

the determinant `| (hati,hatj,hatk), (a_1,a_2,a_3), (b_1,b_2,b_3) |`

`=hati(a_2b_3-a_3b_2)-hatj(a_1b_3-a_3b_1)+hatk(a_1b_2-a_2b_1)`

Here we have,

`〈1.-2-3〉` and `〈2,-5,8〉`

So, the cross product is

`| (hati,hatj,hatk), (1,-2,-3), (2,-5,8) |`

`=hati(-16-15)-hatj(8+6)+hatk(-5+4)`

`=〈-31,-14,-1〉`

Verification (the dot product of perpendicular vectors is `=0`)

`〈-31,-14,-1〉.〈1.-2-3〉=-31+28+3=0`

`〈-31,-14,-1〉.〈2,-5,8〉=-62+70-8=0`