Question

What is the angle between `<7 , 2 , 7 >` and `< 8 , -5 , 0 >`?

Answer 1

`61.132^@`.

Explanation:

The angle `theta` between any 2 vectors `A and B` in `RR^n` may be found using the Euclidean inner product as follows :

`A*B=||A|| ||B|| cos theta`.

`therefore theta = cos^(-1)((A*B)/(||A||||B||))`

`=cos^(-1)((7xx8+2xx-5+7xx0)/(sqrt(7^2+2^2+7^2)sqrt(8^2+5^2+0^2)))`

`=cos^(-1)(46/(sqrt102sqrt89))`

`=61.132^@`.

Alternatively, since these particular vectors are in `RR^3`, we could also have used the vector cross product, where `AxxB=||A||||B||sintheta`.
This will involve a matrix determinant to evaluate and I leave the details as an exercise. It will eventually also give the same final answer.