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Q)

If the projection of $\overrightarrow{a} $ on $\overrightarrow{b} $ and the projection of $\overrightarrow{b}$ on $\overrightarrow{a}$ are equal then the angle between $\overrightarrow{a}+\overrightarrow{b}$ and $\overrightarrow{a}-\overrightarrow{b}$ is

\[\begin{array}{1 1}(1)\frac{\pi}{2} &(2)\frac{\pi}{3}\\(3)\frac{\pi}{4}&(4)\frac{2\pi}{3}\end{array}\]

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A)
Projection of $\overrightarrow{a}$ on $\overrightarrow{b}= \large\frac{\overrightarrow{a}.\overrightarrow{b}}{|\overrightarrow{b}|}$
Projection of $\overrightarrow{b}$ on $\overrightarrow{a}= \large\frac{\overrightarrow{a}.\overrightarrow{b}}{|\overrightarrow{a}|}$
Given $\large\frac{\overrightarrow{a}.\overrightarrow{b}}{|\overrightarrow{b}|} =\large\frac{\overrightarrow{a}.\overrightarrow{b}}{|\overrightarrow{a}|}$
$=>|\overrightarrow{a}|= |\overrightarrow{b}|$
$(\overrightarrow{a}+\overrightarrow{b}).(\overrightarrow{a}-\overrightarrow{b})=\overrightarrow{a}^2- \overrightarrow{b}^2$
$(\overrightarrow{a}+\overrightarrow{b}).(\overrightarrow{a}-\overrightarrow{b})=|\overrightarrow{a}|^2- |\overrightarrow{b}|^2$
$\qquad=0$
$\qquad= \large\frac{\pi}{2} $
Hence 1 is the correct answer.
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