Question

What happens when we multiply a vector by ( $ - 2$ )?
A. Direction reverses and unit changes
B. Direction reverses and magnitude is doubled
C. Direction remains unchanged but unit changes
D. Neither direction reverses nor unit changes but magnitude is doubled

Answer 1

Hint: In order to solve this question we need to understand scalars and vectors. Scalars are those physical quantities which have only magnitude and no direction; however vectors are defined as physical quantities, having both magnitude and direction also it must follow or add to another vector according to triangle law of vector addition. Examples of scalar are, work, energy, power etc. while examples of vectors are force, displacement, velocity.

Complete step by step answer:
Consider a vector $\vec A$ having magnitude equal to “p” and let us say it is in x direction, so it would be represented as, $\vec A = p\hat i$.
Magnitude of vector $\vec A$ is, $\left| {\vec A} \right| = p \to (i)$
Direction of vector $\vec A$ is, $\hat A = \hat i \to (ii)$
Now suppose this vector is multiplied by $ - 2$.Let the new vector be $\vec B$ so according to the question,
$\vec B = - 2\vec A$
Putting the value we get,
$\vec B = - 2(p\hat i)$
So magnitude of $\vec B$ is
$\left| {\vec B} \right| = \left| { - 2p\hat i} \right|$
$\Rightarrow \left| {\vec B} \right| = 2p\left| {\hat i} \right|$
$\Rightarrow \left| {\vec B} \right| = 2p$ Since $\left| {\hat i} \right| = 1$

From equation (i) we get,
$\left| {\vec B} \right| = 2\left| {\vec A} \right|$
So the magnitude of the final vector would be doubled.
For direction of vector $\vec B$ is given by,
\[\hat B = \dfrac{{\vec B}}{{\left| {\vec B} \right|}}\]
Putting values we get,
\[\hat B = \dfrac{{ - 2p\hat i}}{{2p}}\]
\[\Rightarrow \hat B = - \hat i\]
So from equation (ii) we get,
\[\hat B = - \left| {\vec A} \right|\]
So the direction of the final vector is reversed.

So the correct option is B.

Note: It should be remembered that a physical quantity to be considered as vector only when it has both magnitude and direction in addition to it must follow triangle rule of vector addition, if the vector fails to satisfy any one the one criteria, it would not be considered as vector, for example electric current has both magnitude and direction but since it does not follow triangle rule of vector addition, so it cannot be regarded as vector.