Question

What is the angle between `<1 ,3,-8>` and `< 4,1,5>`?

Answer 1

`126,294^@`

Explanation:

There are 2 methods we can use to calculate this algebraically, either using the vector cross product or the vector inner product.
I shall use the latter method as it is quicker and also more general.

The angle between any 2 vectors `A and B` in any dimensional vector space may be given by the inverse cosine of the Euclidean inner product of the 2 vectors divided by the product of the norms of the 2 vectors.
ie. `costheta=(A*B)/(||A||*||B||)`

`therefore theta=cos^(-1) (([(1,3,-8) * (4,1,5)])/(||(1,3,-8)|| * ||(4,1,5)||))`

`=cos^(-1)((4+3-40)/(sqrt(74sqrt42)))`

`=126,294^@`.