Question

The magnitude of scalar and vector products of two vectors are $48\sqrt3$ and 144 respectively. What is the angle between the two vectors?
$\text{A.}\quad \ 30^\circ$
$\text{B.}\quad \ 45^\circ$
$\text{C.}\quad \ 60^\circ$
$\text{D.}\quad \ 90^\circ$

Answer 1

Hint: There are majorly two types of quantities, scalar and vector quantities. All the quantities are divided into these two categories. Scalar quantities are those quantities, which have only magnitude e.g. – mass, speed, pressure, etc. Vector quantities are those which have both magnitude and directions eg – weight, velocity and thrust.

Complete step by step answer:
The scalar product of two vectors is also called the dot product. It’s called a scalar product as its result is a scalar quantity. For example work is scalar but force and displacement is vector. $W = \int {\vec F. d\vec {s}}$
Scalar product of two vectors $\vec A \ and\ \vec B \ is \ \vec A.\vec B =\ |A||B|cos\theta$, where $\theta$ is the angle between the two vectors.
The vector product of two vectors is also called a cross product. It’s called a vector product as its result is a vector quantity. For example the vector product of force and distance from the axis of rotation gives a vector quantity called torque. $\tau = \vec F \times \vec r$.
Vector product of two vectors $\vec A \ and\ \vec B \ is \ \vec A\times \vec B =\ |A||B|sin\theta$ ,where $\theta$ is the angle between the two vectors.
Now, given $|\vec A . \vec B| = |\vec A||\vec B|cos\theta = 48\sqrt 3$ . . . . ①
Also, $|\vec A \times \vec B| = |\vec A||\vec B|sin\theta = 144$ . . . . ②
Dividing equation ② with ①, we get;
$\dfrac{sin\theta}{cos\theta} = \dfrac{144}{48\sqrt 3}$
Or $tan \theta = \sqrt3$
$\implies \theta = 60^\circ$

So, the correct answer is “Option C”.

Note: There are certain vector quantities which are resulted by product of other vector quantities. Like we can’t add vectors by traditional addition (scalar addition), similarly they can’t be multiplied either. Hence we have two types of methods to find the product of vectors, scalar and vector. Scalar product is used if the resultant quantity is a scalar quantity and vector product is used if the resultant quantity is a vector quantity.