Question

The resultant of two vectors of magnitudes $3$ units and $4$ units is $1$ unit. What is the value of their dot product?
A. $ - 12$ units
B. $ - 7$ units
C. $ - 1$ units
D. Zero

Answer 1

Hint: You can start by briefly explaining vector quantities. Then write the equation for the magnitude of the resultant vector, i.e. $|R| = \sqrt {{A^2} + {B^2} + 2AB\cos \theta } $, then use this equation to calculate the value of $\cos \theta $ . Then calculate the value of the dot product of the two vectors by using the equation $\vec R = \vec A \cdot \vec B = AB\cos \theta $.

Complete answer:
We know that the equation of the resultant vector is
$|R| = \sqrt {{A^2} + {B^2} + 2AB\cos \theta } $
Here, $|R| = $ The magnitude of the resultant vector
$A = $ The magnitude of the first vector
$B = $ The magnitude of the second vector
$\theta = $ The angle between the two vectors
So, $1 = \sqrt {{{\left( 3 \right)}^2} + {{\left( 4 \right)}^2} + 2 \times 3 \times 4 \times \cos \theta } $
${\left( 1 \right)^2} = {\left( 3 \right)^2} + {\left( 4 \right)^2} + 2 \times 3 \times 4 \times \cos \theta $
$ - 24 = 24 \times \cos \theta $
$\cos \theta = - 1$ (Equation 1)
We also know that the dot product of the two vectors is given by the equation
$\vec R = \vec A \cdot \vec B = AB\cos \theta $
Let the dot product of these vectors be ${R_D}$ .
So, ${R_D} = 3 \times 4\cos \theta $
${R_D} = 3 \times 4\left( { - 1} \right)$ (From equation 1, $\cos \theta = - 1$ )
${R_D} = - 12$ units

So, the correct answer is “Option A”.

Additional Information:
A vector is a mathematical quantity that has both a magnitude (size) and a direction. To imagine what a vector is like, imagine asking someone for directions in an unknown area and they tell you, “Go $5km$ towards the West”. In this sentence, we see an example of a displacement vector, “ \[5km\] ” is the magnitude of the displacement vector, and “towards the North” is the indicator of the direction of the displacement vector. Some examples of vectors are – Displacement, Force, Acceleration, Velocity, Momentum, etc.

A vector quantity is different from a scalar quantity in the fact that a scalar quantity has only magnitude, but a vector quantity possesses both direction and magnitude. Unlike scalar quantities, vector quantities cannot undergo any mathematical operation, instead, they undergo Dot product and Cross product.

Note:
In the problem given to us we were required to calculated the dot product of the two given vectors, it is quite simple as you saw, but if we were required to calculate the cross product of the two vectors we would have to consider the direction of both the vectors because in normal mathematics $A \times B$ is said to be equal to $B \times A$, but while we are discussing cross product of two vectors $A \times B \ne B \times A$.