Motion in a Plane - Result Question 2
2. 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]
(a) $0^{\circ}$
(b) $90^{\circ}$
(c) $45^{\circ}$
(d) $180^{\circ}$
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Answer:
Correct Answer: 2. (b)
Solution:
- (b) $|\overrightarrow{{}A}+\overrightarrow{{}B}|=|\overrightarrow{{}A}-\overrightarrow{{}B}|$
Squaring on both sides
$ \begin{aligned} & |\overrightarrow{{}A}+\overrightarrow{{}B}|^{2}=|\overrightarrow{{}A}-\overrightarrow{{}B}|^{2} \\ & \Rightarrow \overrightarrow{{}A} \cdot \overrightarrow{{}A}+2 \overrightarrow{{}A} \cdot \overrightarrow{{}B}+\overrightarrow{{}B} \cdot \overrightarrow{{}B} \\ & \quad=\overrightarrow{{}A} \cdot \overrightarrow{{}A}-2 \overrightarrow{{}A} \cdot \overrightarrow{{}B}+\overrightarrow{{}B} \cdot \overrightarrow{{}B} \\ & \Rightarrow 4 \overrightarrow{{}A} \cdot \overrightarrow{{}B}=0 \Rightarrow 4 AB \cos \theta=0 \\ & \Rightarrow \cos \theta=0 \Rightarrow \theta=90^{\circ} \end{aligned} $
