Question

If for two vector ,$\overrightarrow{A}$ and $\overrightarrow{B}$, sum $(\overrightarrow{A}+\overrightarrow{B})$ is perpendicular to the difference $(\overrightarrow{A}-\overrightarrow{B})$. The ratio of their magnitude is

• A. $1$
• B. $2$
• C. $3$
• D. none of these

Answer 1

The correct option is A $1$

$(\overrightarrow{A}+\overrightarrow{B})\text{is perpendicular to}(\overrightarrow{A}-\overrightarrow{B})$
Thus, $(\overrightarrow{A}+\overrightarrow{B}).(\overrightarrow{A}-\overrightarrow{B})=0$

Commutative property of dot product $\overrightarrow{A}.\overrightarrow{B}=\overrightarrow{B}.\overrightarrow{A}$
$\therefore A^2-\overrightarrow{A}.\overrightarrow{B}+\overrightarrow{B}.\overrightarrow{A}-B^2=0A^2-B^2=0\Rightarrow A=\pm B$

Thus the ratio of magnitudes:
$\frac{A}{B}=1$ (since , magnitude cannot be negative)