Question

The magnitude of the sum of the two vectors is equal to the difference of their magnitudes, the angle between the vectors is,

• A. 0$^\circ$
• B. 45$^\circ$
• C. 90$^\circ$
• D. 180$^\circ$

Answer 1

Answer: (C)

90$^\circ$

Answer 2

Let the two vectors be $\vec{A}$ and $\vec{B}$ with magnitudes $A$ and $B$ respectively. Then magnitude of their sum is given by:
$|\overrightarrow{ A }+\overrightarrow{ B }|=\sqrt{ A ^{2}+ B ^{2}+2 AB \cos \theta} \rightarrow(1)(\theta=$ angle between the vectors $)$
Magnitude of their difference is given by:
$|\overrightarrow{ A }-\overrightarrow{ B }|=\sqrt{ A ^{2}+ B ^{2}-2 AB \cos \theta} \rightarrow(2)$
As $|\overrightarrow{ A }+\overrightarrow{ B }|=|\overrightarrow{ A }-\overrightarrow{ B }|$
$\Rightarrow A ^{2}+ B ^{2}+2 AB \cos \theta= A ^{2}+ B ^{2}-2 AB \cos \theta$
$\Rightarrow 4 AB \cos \theta=0$ or $\cos \theta=0$
$\Rightarrow \theta=90^{\circ} .$