Question

if |vector a+ vector b| = |vector a- vector b|find the angle between a and b

Answer 1

let A+B=C, and c represent the magnatude of vectorC, or |vector a+ vector b| =c

then law of cosines
c^2=a^2+b^2-2ABcosAngleAB
Now consider |vector a- vector b|. The angle between A and B now is the supplement to the first angle AB. We can call this 180-AB.

law of cosines
c^2=a^2+b^2-2ABcosAngle(180-AB)
take the two equations, and subtract them.

0=0+0-2abcosAB -2abcos(180-AB)
or cosAB=-cos(180-AB)
cosAB= Cos180CosAB+sin180sinAB
cosAB= -cosAB+0
or 2cosAB=0 which means AB=90 deg.