Question

What is the dot product of `<8,-1,4>` and `<9,1,5 >`?

Answer 1

`91`

Explanation:

For any 2 vectors `v=(v_1,v_2,....,v_n) and w=(w_1,w_2,...,w_n)` in a real or complex finite dimensional vector space, the Euclidean inner product (dot product) is defined as follows :

`v*w=v_1w_1+v_2w_2+.....+v_nw_n`.

So in this particular case of 3-dimesnional vectors we get

`(8,-1,4)*(9,1,5)=(8xx9)+(-1xx1)+(4xx5)`

`=72-1+20`

`=91`.

(Note that we can also represent the inner product as the angle between the 2 vectors and the norms of the 2 vectors as follows :
`u*v=||u||xx||v||costheta`