Question

If `A= <3, -1 >` and `B= <-8, 6>`, what is `||A+B|| -||A|| -||B||`?

Answer 1

We find the length of the resultant vector from adding two vectors together and then subtract from it the length of the two component vectors.

`||A+B||-||A||-||B||=7.07-3.16-10=-6.09`

Explanation:

We are effectively finding the length of the new vectors that is the sum of two vectors, minus the lengths of each of the component vectors.

If you draw a diagram you will see why we expect the answer to be negative: the resultant vector is shorter than the two vectors that make it up (unless those two vectors are in the same direction).

First step is to add the vectors:

`A+B = <3,-1>`+`<-8,6> = <-5,5>`

Second is to find the lengths of each of the three vectors we're interested in: `A, B` and `A+B`.

`||A||=sqrt(3^2+(-1)^2)=sqrt10=3.16`

`||B||=sqrt((-8)^2+6^2)=sqrt100=10`

`||A+B||=sqrt((-5)^2+5^2)=sqrt50=7.07`

Pulling it all together,

`||A+B||-||A||-||B||=7.07-3.16-10=-6.09`

As we predicted, the answer is negative.