Question

If |A+B| = |A-B| , then what is the angle between A and B ?

Answer 1

If A+B=A-B what is the angle between A and B

I assume the question means |A+B|=|A-B| where A and B are non-zero vectors.

Squaring both sides,

|A+B|2=|A−B|2|A+B|2=|A−B|2

Since A.A=|A|2A.A=|A|2

|A+B|2=|A−B|2|A+B|2=|A−B|2

(A+B).(A+B)=(A−B).(A−B)(A+B).(A+B)=(A−B).(A−B)

A.A+A.B+B.A+B.B=A.A−A.B−B.A+B.BA.A+A.B+B.A+B.B=A.A−A.B−B.A+B.B( Using distributive property)

|A|2+2A.B+|B|2=|A|2−2A.B+|B|2|A|2+2A.B+|B|2=|A|2−2A.B+|B|2

4A.B=04A.B=0

A.B=0A.B=0

|A||B|cos(theta)=0|A||B|cos(theta)=0

Since A and B are non-zero vectors, A and B must be perpendicular as cos (90 degrees) = 0