Question

Two vectors both are equal in magnitude and have their resultant equal in magnitude of the either. Find the angle between the two vectors.

Answer 1

Here we say we have two vectors A and B. It is given that the magnitude of both the vectors is the same and also the resultant vector comes out to be having the same magnitude. We know for addition of vectors we can’t use the ordinary laws of Algebra. To add two or more than two vectors we have to use either the triangle law of vector addition or parallelogram law of vector addition.

Complete step by step answer:
Let the magnitude of both the vectors and the resultant be x.
$\therefore \left| A \right|=x=\left| B \right|=\left| R \right|$
By using the parallelogram law, we can find the angle.
$R=\sqrt{A{}^\text{2}+B{}^\text{2}+2AB\cos \varphi }$ , where A and B are two forces, R is their resultant and $\varphi$ is the angle between them.
$R=\sqrt{A{}^\text{2}+B{}^\text{2}+2AB\cos \varphi } \\\ \Rightarrow x=\sqrt{x{}^\text{2}+x{}^\text{2}+2{{x}^{2}}\cos \varphi } \\\ \Rightarrow {{x}^{2}}=x{}^\text{2}+x{}^\text{2}+2{{x}^{2}}\cos \varphi \\\ \Rightarrow {{x}^{2}}=2{{x}^{2}}(1+\cos \varphi ) \\\ \Rightarrow 0.5=1+\cos \varphi \\\ \Rightarrow \cos \varphi =-0.5 \\\ \Rightarrow \varphi ={{\cos }^{-1}}(-0.5) \\\ \therefore \varphi ={{120}^{0}}$

So, the angle comes out to be ${{120}^{0}}$.

Additional Information:
Most of the time we get confused that if the magnitude is zero then the given quantity cannot be termed as a vector but in actual it depends upon the state of the quantity. If the distance between two points is zero then that cannot be called a zero vector because distance is not a vector quantity but that holds true for displacement

Note: Even triangle law can be applied here, where the forces F and F acting at a point are shown based on magnitude and direction, by the two sides of a triangle taken in an order, and the obtained resultant is shown by the third side of the triangle which is taken in the opposite order. We have used parallelogram law where the two vectors acting at a point are shown based on magnitude and direction, by the two adjacent sides of a parallelogram generated from a particular point and the obtained resultant is represented by the diagonal of the parallelogram.