Question

-A) The projection of a vector on another vector gives the component along that vector.
B) The component of a vector is a scalar and can be converted to a vector component by multiplying with a unit vector.
1- Both A and B are correct
2- Both A and B are wrong
3- A is correct but B is wrong
4- B is correct but A is wrong

Answer 1

Hint: While finding projection of one vector over another we need to consider the dot product. There are two ways of multiplication of a vector with another and one is dot product while other is cross product. Vectors have components which can be found by resolving a vector.

Complete step by step solution:
Suppose we have two vectors \[\overrightarrow{A}\And \overrightarrow{B}\]
Dot product or scalar product of two vectors is given by \[\overrightarrow{A}.\overrightarrow{B}=AB\cos \alpha \]
 \[\alpha \]is the angle between the two vectors. Now Projection of \[\overrightarrow{A}\]on \[\overrightarrow{B}\]is given as: \[\left| A \right|\dfrac{\overrightarrow{A}.\overrightarrow{B}}{\left| A \right|\left| B \right|}\]
Now we can see that since, the numerator contains dot product which always gives a scalar quantity and in the denominator is also scalar, thus, overall projection gives scalar result. Thus, A is correct.
If we multiply a scalar say \[\alpha \]with any vector then the result is a vector. Also, the component of a vector is found by resolving it and resolving a vector always gives a scalar number, thus B is also correct.

So, the correct answer is “Option 1”.

Note:
While taking either dot product or cross product we have to keep in mind we have to take the angle between the two original vectors. While resolving a vector we need to have a particular directional axis say X axis or Y axis.