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:
(A) ${180^\circ }$
(B) ${0^\circ }$
(C) ${90^\circ }$
(D) ${45^\circ }$

Answer 1

In order to answer this question, first we will assume the two variables, and then we will write the magnitude of sum of two vectors and then magnitude of difference of two vectors. And then we will equate both of them as per the question to find the dot product of the equation, as we will know the angle between both the vectors.

Complete step by step solution:
Let the two vectors be .
So, sum of two vectors is, ,
And, the magnitude of sum of two vectors is, .
Now,
Difference of two vectors is, ,
And, the magnitude of difference of two vectors is, .
Now, according to the question, the magnitude of sum of two vectors is equal to the magnitude of difference of the two vectors:
Now, we have to find the angle between the both vectors, we will squaring the both sides of upper equation:-
Now, we will cancel out ${A^2} + {B^2}$ from both the sides:
Therefore, as dot products of the two vectors are zero, the angle between them will be ${90^\circ }$ .
Hence, the correct option is (C) ${90^\circ }$ .

Note:
The shortest angle between two vectors, deferred by a single point, is the angle at which one of the vectors must be turned around to be co-directional with another vector. The most fundamental relationship. The dot product of these vectors divided by the product of vector magnitude equals the cosine of the angle between two vectors.