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

Feb 15, 2016

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 $A + B$.

$| | A | | = \sqrt{{3}^{2} + {\left(- 1\right)}^{2}} = \sqrt{10} = 3.16$

$| | B | | = \sqrt{{\left(- 8\right)}^{2} + {6}^{2}} = \sqrt{100} = 10$

$| | A + B | | = \sqrt{{\left(- 5\right)}^{2} + {5}^{2}} = \sqrt{50} = 7.07$

Pulling it all together,

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

As we predicted, the answer is negative.