Question

If `A= <2 , 6 >` and `B= <-1,7 >`, what is `||A+B|| -||A|| -||B||`?

Answer 1

In words, we're asked to add two vectors, then find the length of the combined vector and subtract from it the length of each of the individual vectors. `||A+B||-||A||-||B||=13-6.3-7.1=-0.4`

Explanation:

Step 1 - add the vectors:

`A+B=<2,6>+<-1,7>` `=<1,13>`

Step 2 - find the length of the new vector:

`||A+B||=sqrt(1^2+13^2)=sqrt170=13`

Step 3: find the lengths of `A` and `B`:

`||A||=sqrt(2^2+6^2)=sqrt(4+36)=sqrt40=6.3`

`||B||=sqrt(-1^2+7^2)=sqrt(1+49)=sqrt50=7.1`

Step 4: calculate the answer:

`||A+B||-||A||-||B||=13-6.3-7.1=-0.4`

This makes sense, since the vectors are not in the same direction as each other. The length of the vector sum is a little shorter than the sum of the lengths of the original vectors.