Question

If `A= <-5 ,-4 ,-7 >` and `B= <2 ,-9 ,8 >`, what is `A*B -||A|| ||B||`?

Answer 1

`-13260`

Explanation:

To answer this question we need three pieces:

• `A*B`
• `||A||`
• `||B||`

but don't worry, they're all easy to compute!

• `A*B` is the scalar product of two vectors. As the name suggests, it is a number (scalar), computed as follows: `A` and `B` must have the same length, and we must add the products of the coordinates of `A` and `B` in the same positions.

So, if `A=(a_1,a_2,...,a_n)` and `B=(b_1,b_2,...,b_n)`, we have

`A*B = a_1b_1+a_2b_2+...+a_nb_n = sum_{i=1}^n a_ib_i`

So, in your case, we have

`A*B = -5*2 + (-4)(-9) + (-7)*8 = -10+36-56=-30`

The other pieces, `||A||` and `||B||`, are called the norm of `A` and `B`. Actually, the norm is defined as the scalar product of a vector with itself: `||A|| = A*A`, we already know how to compute them:

`||A|| = (-5)(-5)+(-4)(-4)+(-7)(-7) = (-5)^2+(-4)^2+(-7)^2 = 25+16+49=90`

`||B|| = 2^2+(-9)^2+8^2 = 2+81+64=147`

We're finally ready!

`A*B - ||A|||B|| = -30 - 90*147 = -30-13230 = -13260`