Suppose that A and B are vectors so that A+2B = A-2B have the same magnitude. Why do A and B have to be orthogonal?

Question

Suppose that A and B are vectors so that A+2B and A-2B have the same magnitude. Explain why A and B are orthogonal.

Answer 1

Hint: $\lVert\mathbf{A}+2\mathbf{B}\rVert^2=(\mathbf{A}+2\mathbf{B})\cdot(\mathbf{A}+2\mathbf{B})=\ldots$

Answer 2

Use the definition of norm in terms of dot product and cancel similar terms to get the result.

Answer 3

$A+2B$ and $A-2B$ are diagonals of a parallelogram with sides $|A|$ and $|2B|.$ Since the diagonals are equal, it is a rectangle. Also the vectors A and B are orthogonal.