Question

The sum of two vectors $A$ and $B$ is at right angle to their difference. Then,

• A. $A=B$
• B. $A=2B$
• C. $B=2A$
• D. $A$ and $B$ have the same direction

Answer 1

The correct option is A

$A=B$

Explanation for the correct option:

Let $r_1$ and $r_2$ be the sum and difference of vectors $A$ and $B$ respectively, i.e.,

$$\begin{gathered} r_1=A+B \\ {r}_2=A-B \end{gathered}$$

Given, $r_1$ is perpendicular to $r_2$.

Taking the dot product of $r_1$ and $r_2$,

$\begin{array}{ccc}& r_1·r_2=\left(A+B\right)\left(A-B\right)& \\ \Rightarrow & 0=A^2-B^2& \left[\because r_1·r_2=0\mathrm{if}{r}_1\perp r_2\right]\\ \Rightarrow & A^2=B^2& \\ \Rightarrow & A=B& \end{array}$

Hence,$\mathrm{Option}\left(\mathrm{A}\right)$ is correct.