Question

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:
\[
  {\text{A}}{\text{. 0}}^\circ \\
  {\text{B}}{\text{. 90}}^\circ \\
  {\text{C}}{\text{. 45}}^\circ \\
  {\text{D}}{\text{. 180}}^\circ \\
 \]

Answer 1

Hint: First, we need to assume two vectors with a certain angle between them and then find the magnitude of their sum and their difference. Then on equating their magnitudes and solving for the value of angle, we can get the required answer.

Complete step-by-step answer:
We are given that the magnitude of the sum of two vectors is equal to the magnitude of difference of the two vectors. Let us consider $\overrightarrow A $ and $\overrightarrow B $ to be the two vectors which satisfy the given condition.
Now let us first write the magnitude of the sum of these two vectors which can be written in the following way.
$|\overrightarrow A + \overrightarrow B | = \sqrt {{A^2} + {B^2} + 2AB\cos \theta } $ …(i)
Here angle $\theta $ represents the angle between vectors A and B which we are required to find out.
Similarly, we can write the magnitude of difference of these two vectors which can be given in the following way.
$|\overrightarrow A - \overrightarrow B | = \sqrt {{A^2} + {B^2} - 2AB\cos \theta } $ …(ii)
According to the given condition, the magnitude of sum of two vectors is equal to the magnitude of difference of the two vectors. Therefore, we can write
$|\overrightarrow A + \overrightarrow B | = |\overrightarrow A - \overrightarrow B |$
Now we can insert the expressions from equation (i) and (ii) in this equation. Doing so we get
$\sqrt {{A^2} + {B^2} + 2AB\cos \theta } = \sqrt {{A^2} + {B^2} - 2AB\cos \theta } $
Now on squaring both sides of this equation, we get the following expression.
$
  {A^2} + {B^2} + 2AB\cos \theta = {A^2} + {B^2} - 2AB\cos \theta \\
   \Rightarrow 4AB\cos \theta = 0 \\
   \Rightarrow \cos \theta = 0 \\
   \Rightarrow \theta = 90^\circ \\
 $
Hence, the angle between the two given vectors is 90$^\circ $. So, the correct answer is option B.

So, the correct answer is “Option B”.

Note: The magnitude of a vector represents the length of the vector. On adding and subtracting two vectors, we get two different vectors and in the question, the lengths of these two new vectors were equal from which we have obtained the angle between the two original vectors by using simple mathematics.