A question on Vector addition.

Question

Is (A vector +B vector) always perpendicular to (A vector - B vector)?

Answer 1

No, of course not. Why do you think it would? Try almost any two random vectors in the basic spaces $\;\Bbb R^n\;$ , for example, with the usual inner product:

$$a:=\binom10,\,\,b:=\binom11\implies a+b=\binom21\;,\;\;a-b=\binom0{-1}$$

Answer 2

Try to get resultant from the parallelogram law, and you will see that $(\vec A + \vec B)$ and $(\vec A - \vec B)$ are perpendicular only when their magnitude is equal.

Resultant of $(\vec A + \vec B)=\sqrt{a^2+b^2+2AB\cos\theta}$ and resultant of $(\vec A - \vec B)=\sqrt{a^2+b^2-2AB\cos\theta}$ where $\theta$ is the angle between $\vec A $ and $\vec B$.

Answer 3

No.

Let $a, b$ be two vectors in a vector space equipped with an inner product, the necessary condition would be: $\left\|a\right\| = \left\|b\right\|$.

The derivation is

\begin{align} (a + b)(a - b) = a*a - b*b = \left\|a\right\| - \left\|b\right\| = 0 \end{align}