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

1 Answer
David G.
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||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>A+B=<3,−1>+<-8,6> = <-5,5><−8,6>=<−5,5>

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

||A||=sqrt(3^2+(-1)^2)=sqrt10=3.16||A||=√32+(−1)2=√10=3.16

||B||=sqrt((-8)^2+6^2)=sqrt100=10||B||=√(−8)2+62=√100=10

||A+B||=sqrt((-5)^2+5^2)=sqrt50=7.07||A+B||=√(−5)2+52=√50=7.07

Pulling it all together,

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

As we predicted, the answer is negative.

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