Question

If `A= <6 ,9 ,-1 >` and `B= <-4 ,-1 ,4 >`, what is `A*B -||A|| ||B||`?

Answer 1

`62.401`.

Explanation:

Definitions : Let `A=(a_1,a_2,....,a_n) and B=(b_1,b_2,....,b_n)` be any 2 vectors in a real or complex finite dimensional vector space X. Then we define:

1. The Euclidean inner product (dot product) of A and B as the real or complex number given by `A*B= a_1b_1+a_2b_2+......+a_nb_n`.
2. The norm of A as the real or complex number given by `||A||=sqrt(a_1^2+a_2^2+......+a_n^2)`.

Applying these 2 definitions to the given 3 dimensional vectors we get :

`A*B=(6,9,-1)*(-4,-1,4)`

`=(6xx-4)+(9xx-1)+(-1xx4)`

`=-24-9-4`

`=-37`.

`||A||=|| (6,9,-1) || = sqrt(6^2+9^2+1^2)=sqrt118`.

Similarly `||B||= sqrt33`.

`therefore A*B-||A|| ||B|| = -37-(sqrt118)(sqrt33)`

`=-37-sqrt(118xx33)`

`=62.401`.