First slide

Vectors

Question

If | $\vec{A} + \vec{B}$ | = | $\vec{A} - \vec{B}$ |, then the angle between $\vec{A} and \vec{B}$ will be

Moderate

A

$30^{0}$
B

$45^{0}$
C

$60^{0}$
D

$90^{0}$

Solution

Let $θ$ be the angle between the vectors $\vec{A} and \vec{B}$ .
Then
| $\vec{A} + \vec{B}$ | = $\sqrt{A^{2} + B^{2} + 2 ABcosθ}$
| $\vec{A} - \vec{B}$ | = $\sqrt{A^{2} + B^{2} - 2 ABcosθ}$
| $\vec{A} + \vec{B}$ | = | $\vec{A} - \vec{B}$ |
$∴$ $\sqrt{A^{2} + B^{2} + 2 ABcosθ}$ = $\sqrt{A^{2} + B^{2} - 2 ABcosθ}$
Squaring on both sides, we get
$A^{2} + B^{2} + 2 ABcosθ = A^{2} + B^{2} - 2 ABcosθ$
4AB $cosθ$ = 0
$∴$ $cosθ$ = 0 or $θ = \frac{π}{2}$

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