## OR: Can the sum of two equal vectors be equal to either of the vectors?

## OR: Two vectors have equal magnitude and their resultant also has the same magnitude. What is the angle between the two vectors?

## OR: Under what condition will the sum of two vectors of equal magnitude have magnitude equal to either vector?

Let the two vectors be $\overrightarrow{P}$ and $\overrightarrow{Q}$ and let $R$ be their resultant. If $\theta$ is the angle between the two vectors, then,

\[R=\sqrt{P^2+Q^2+2PQ\cos\theta}\]

[**From:** Parallelogram Law of Vector Addition]

According to the question, $R=P=Q$.

\[\therefore R=\sqrt{R^2+R^2+2R^2\cos\theta}\] \[R=R\sqrt{2+2\cos\theta}\] \[1=2(1+\cos\theta)\] \[\frac{1}{2}=1+\cos\theta\] \[\cos\theta=-\frac{1}{2}\] \[\therefore\theta=120°\]

Hence, the angle between the two vectors should be $120°$.