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

1 Answer
David G.
Feb 11, 2016

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.

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