Question

If the magnitude of the sum of two vectors is equal to the magnitude of the difference of the two vectors, then the angle between these vectors is

• A. $0°$
• B. $45°$
• C. $90°$
• D. $180°$

Answer 1

The correct option is C

$90°$

Step 1. Given data:

1. Let two vectors be $P$ and $Q$.
2. It is given that $\left|\begin{array}{c}P+Q\end{array}\right|=\left|\begin{array}{c}P-Q\end{array}\right|$…….$\left(1\right)$

Step 2. Calculations:

1. Let angle between $P$ and $Q$ is $\phi$.
2. Squaring and rearranging $\left(1\right)$ both sides, we get

$$\begin{gathered} P^2+Q^2+2PQ.\cos \left(\phi \right)=P^2+Q^2-2PQ\cos \left(\phi \right) \\ 4PQ\cos \left(\phi \right)=0 \\ \cos \left(\phi \right)=0 \\ \phi ={\cos }^{-1}\left(0\right)=90° \end{gathered}$$

Thus, the angle between these vectors is $90°$.