Question

Assertion :If $|\overrightarrow{a}+\overrightarrow{b}|=|\overrightarrow{a}-\overrightarrow{b}|$, then $\overrightarrow{a}$ and $\overrightarrow{b}$ are perpendicular to each other Reason: If the diagonals of a parallelogram are equal in magnitude, then the parallelogram is a rectangle.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two vectors
Hence
$|\overrightarrow{a}+\overrightarrow{b}|=|\overrightarrow{a}-\overrightarrow{b}|$

$\sqrt{a^2+b^2+2ab\cos \theta }=\sqrt{a^2+b^2-2ab\cos \theta }$

Squaring on both the sides, we get
$a^2+b^2+2ab\cos \theta =a^2+b^2-2ab\cos \theta$
$4ab\cos \theta =0$ ...$\left(i\right)$

If $a\ne 0$ $b\ne 0$
Then $\cos \theta =0$
$\theta =\frac{(2n-1)\pi }{2}$ where $nϵN$

Hence $\overrightarrow{a}$ is perpendicular to $\overrightarrow{b}$.
If $\overrightarrow{a}$ and $\overrightarrow{b}$ are adjacent sides of a parallelogram, then, the parallelogram is a rectangle. ...(from $i$).