Question

If \[\overrightarrow{a}\] is a non- zero vector of modulus a and m is a non-zero scalar, then \[m\overrightarrow{a}\] is a unit vector if

Answer 1

Hint: According to Newton’s second law of motion force is the product of mass and acceleration. Since the product m and a are unity. By rearranging them we will get the relation connecting m and \[\overrightarrow{a}\]. When we multiply a scalar with a vector quantity the resultant quantity will be a vector.

Complete step-by-step solution:
If we multiply a scalar with a vector quantity the resultant quantity will be a vector. Here the scalar quantity is mass and acceleration is a vector quantity. Here the vector quantity is called force.
Given that \[\overrightarrow{a}\] is a non- zero vector of modulus a and m is a non-zero scalar. Also given that product of m and a is unity. That is,
\[\left| m\overrightarrow{a} \right|=1\]
The product of mass and acceleration \[m\overrightarrow{a}\]is a unit vector only if,
\[m=\dfrac{1}{\left| \overrightarrow{a} \right|}\]
Here m becomes the reciprocal of \[\overrightarrow{a}\].
Where, \[\overrightarrow{a}\] is a non- zero vector of modulus a and m is a non-zero scalar.
Thus the answer is \[m=\dfrac{1}{\left| \overrightarrow{a} \right|}\].

Note: If we multiply a scalar with a vector quantity the resultant quantity will be a vector. Here the scalar quantity is mass and acceleration is a vector quantity. So while taking the product of those two quantities the resultant quantity is a vector. Here the vector quantity is called force. That is the product of mass and acceleration gives the force which is Newton’s second law of motion. Here the force will be unity only if m becomes the reciprocal \[\overrightarrow{a}\].