If the magnitude of sum of two vectors is equal to the magnitude of difference of the two vectors, the angle between these vectors is [2016]

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Solution

Q.
If the magnitude of sum of two vectors is equal to the magnitude of difference of the two vectors, the angle between these vectors is [2016]

1 $45 °$

2 $180 °$

3 $0 °$

4 $90 °$

Ans.
(4)

Let the two vectors be $\vec{A}$ and $\vec{B}$ .

Then, magnitude of sum of $\vec{A}$ and $\vec{B},$ $| \vec{A} + \vec{B} |$ $= \sqrt{A^{2} + B^{2} + 2 A B \cos θ}$

and magnitude of difference of $\vec{A}$ and $\vec{B}$

$| \vec{A} - \vec{B} |$ $= \sqrt{A^{2} + B^{2} - 2 A B \cos θ}, | \vec{A} + \vec{B} | = | \vec{A} - \vec{B} | (given)$

or $\sqrt{A^{2} + B^{2} + 2 A B \cos θ} = \sqrt{A^{2} + B^{2} - 2 A B \cos θ}$

$\Rightarrow 4 A B \cos θ = 0$ $∵$ $4 A B \neq 0,$ $∴$ $\cos θ = 0 or θ = 90 °$

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