Find $ |a+b|$ knowing $|a|=4, |b|=6, |a-b|=5$

Question

How to find $|a+b|$ knowing $|a|=4, |b|=6, |a-b|=5$, where $a, b$ are vectors? I know how to do this using the parallelogram law or the law of cosines, but is there a way to do this algebraically?

Answer 1

Hint: Here is the parallelogram law in inner product spaces: $2\|x\|^2+2\|y\|^2=\|x+y\|^2+\|x-y\|^2$

Answer 2

Here's an algebraic approach: (Note '$\cdot$' denotes dot product.)

$| a-b |^2=(a-b)\cdot (a-b)=|a|^2-2a\cdot b+|b|^2$

From this and the given information you can determine $a\cdot b$.

Then use $| a+b |^2=(a+b)\cdot (a+b)=...$ to find $|a+b|$