Question

A vector is not going to be change if
A. It is displaced parallel to itself.
B. It is rotated arbitrarily.
C. It is cross-multiplied by a unit vector.
D. It is multiplied by an arbitrary scalar.

Answer 1

Hint: In this question, we ask about the conditions for which a vector does not change. As we know that the vector quantities are those who are having magnitude as well as the directions. So, we have to take the condition one by one and see whether the vector is changing or not.

Complete step-by-step answer :
Let us take each condition one by one and see whether the vector is changing or not.
In option A), it is given that the vector remains the same when it is displaced parallel to itself. Now, if we take a vector and displace it in the parallel position from taking it from there initial position to the final position. It will not be going to be changed. So, option A) is correct that the vector is not changing.

In option B), it is given that when the vector is rotated through an arbitrary. When the vector is rotating at some angle it will change the direction of the vector. Therefore, Option B) is not correct.

In option C), it is given that when we cross multiplied by a unit vector. If we cross multiply the vector by a unit vector it will change the vector notation. For example, let us suppose a vector $\vec A$ and it is along with $\mathop i\limits^ \wedge $direction and the normal is on $\mathop j\limits^ \wedge $ direction when we take the cross multiplication we get,
$\vec A\mathop i\limits^ \wedge \times \mathop j\limits^ \wedge \\
\Rightarrow \vec A\mathop k\limits^ \wedge \\ $
Therefore, we can see that it changes the vector direction. Therefore, option C) is not the correct answer.

In option D), it is given that we multiplied by an arbitrary scalar. If a vector is multiplied by a scalar then its quantity will be going to increase, therefore, it will not become the same as itself. Therefore, Option D) is not the correct answer.
By analyzing all the given options, we found out that Option A) is satisfying in that the vector is not changing.

Hence, Option A is the correct answer. .

Note: Do not confuse vector quantity and scalar quantities. Always use the cross product in a proper format. Do not confuse the cross product with the dot product.