Question

What is the angle between vectors $2a$ and $4a$? What is the angle between vectors $3a$ and $-5a$?

Answer 1

The context of this question is the vector and scalar concepts and so the angle between the given vectors can only be answered with the help of the concept of the scalar multiplication of vectors.

Complete answer:
The resultant vector formed when a vector quantity is multiplied by a scalar quantity is the scalar quantity multiplied by the magnitude of the vector. However, the direction is the same. And in the question, the vectors have the magnitude of $2a$ and $4a$ and so the vectors are $2\overrightarrow{a}$ and $4\overrightarrow{a}$. Hence, the direction of both the vectors is $\overrightarrow{a}$ and so they are collinear as they have the same direction and different magnitude. So, the answer to the first part of the question is zero as the vectors are collinear.

When we look at the second part of the question, the magnitude of two vectors are $3a$ and $-5a$ and so using the scalar multiplication of vectors, the two vectors are $3\overrightarrow{a}$ and $-5\overrightarrow{a}$. So, one vector is antiparallel to the other as one of the vectors has a positive direction and the other has a negative direction, so when the vectors are completely opposite in direction, the angle between the vectors is $180{}^\circ$.

Note: In many places, the answer to this question is given with the help of the vector multiplication formula where we can easily find out the angle between the vectors with the help of the magnitude of the vectors.