Question

If the magnitude of vectors \[\overrightarrow A \], \[\overrightarrow B \] and \[\overrightarrow C \] are 12, 5 and 13, respectively and \[\overrightarrow C = \overrightarrow A + \overrightarrow B \], then what the angle between the vectors \[\overrightarrow A \] and \[\overrightarrow B \]?
A. \[\dfrac{\pi }{4}\]
B. \[\dfrac{\pi }{3}\]
C. \[\dfrac{\pi }{2}\]
D. Zero

Answer 1

Hint:Before we proceed with the problem let’s see about vectors and what data they have given and how to solve this question. The quantity which has both magnitude and direction is known as a vector. The magnitude of vectors \[\overrightarrow A \], \[\overrightarrow B \] and \[\overrightarrow C \]are given and we need to find the angle between \[\overrightarrow A \] and \[\overrightarrow B \]. To find this first we need to write the formula to find the magnitude of vectors, then we will find the angle between \[\overrightarrow A \]and \[\overrightarrow B \].

Formula Used:
The formula to find the magnitude of two vectors is given by,
\[{\left| C \right|^2} = {\left| A \right|^2} + {\left| B \right|^2} + 2\left| A \right|\left| B \right|\cos \theta \]……… (1)
Where, \[\theta \] is the angle between vectors \[\overrightarrow A \] and \[\overrightarrow B \].

Complete step by step solution:
Here, the magnitude of vectors of \[\overrightarrow A \], \[\overrightarrow B \] and \[\overrightarrow C \] are given as 12, 5 and 13 respectively. Given that,
\[\overrightarrow C = \overrightarrow A + \overrightarrow B \]
In order to find the angle between the vectors \[\overrightarrow A \] and \[\overrightarrow B \] we need to find and write a formula for the magnitudes of vectors. That is, their magnitudes are related as shown in the equation
\[{\left| C \right|^2} = {\left| A \right|^2} + {\left| B \right|^2} + 2\left| A \right|\left| B \right|\cos \theta \]
Here, \[\theta \] is the angle between vectors \[\overrightarrow A \] and \[\overrightarrow B \].

Therefore,
\[{13^2} = {12^2} + {5^2} + 2 \times \left( {12 \times 5} \right)\cos \theta \]
\[\Rightarrow 169 = 144 + 25 + 120\cos \theta \]
\[\Rightarrow 169 = 169 + 120\cos \theta \]
\[\Rightarrow 120\cos \theta = 0\]
\[ \Rightarrow \cos \theta = 0\]
\[\theta = {90^0}\]
In radians, it can be written as,
\[\therefore \theta = \dfrac{\pi }{2}\]
Therefore, the angle between the two vectors \[\overrightarrow A \] and \[\overrightarrow B \] is \[\dfrac{\pi }{2}\].

Hence, option C is the correct answer.

Note:In order to give a few examples for vectors, any quantity which has both magnitude and direction can be used as an example. Velocity, acceleration and the forces acting on an object can be described using vectors. The magnitude of a vector is the length of the vector.