Question

If vector \[a + b = \] vector \[c\]and \[a + b = c\] what is the angle between vector \[a\] and \[b\]\[?\]
A. \[90\]
B. \[45\]
C. \[0\]
D. \[60\]

Answer 1

Hint: First we have to know that a vector is the physical quantity that has both magnitude and direction. Some examples of vector quantities are velocity, force, and torque.
To find the angle between two vectors we have to know the dot product of two vectors (i.e., \[\vec a.\vec b = \left| {\vec a} \right|\left| {\vec b} \right|\cos \theta \] where \[\theta \] is the angle between two vectors \[\vec a\] and \[\vec b\]) and it is the easiest way. Then using the given condition and dot product we find the angle between given vectors.

Complete step by step answer:
The angle \[\theta \] between two vectors \[\vec a\] and \[\vec b\] is given by
\[\cos \theta = \dfrac{{\vec a.\vec b}}{{\left| {\vec a} \right|\left| {\vec b} \right|}}\]----(1)

Given \[\vec a + \vec b = \vec c\]---(2) and  \[a + b = c\]---(3)
Where \[\left| {\vec a} \right| = a\], \[\left| {\vec b} \right| = b\], and \[\left| {\vec c} \right| = c\].
Squaring both sides of the equation (3), we get
\[{\left( {a + b} \right)^2} = {c^2}\]
\[ \Rightarrow {c^2} = {\left( {a + b} \right)^2}\]----(4)
Since we know that \[{\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab\], then the equation (3) becomes
\[{c^2} = {a^2} + {b^2} + 2ab\]---(5)
Also,
\[\vec c.\vec c = \left( {\vec a + \vec b} \right).\left( {\vec a + \vec b} \right) = \vec a.\vec a + \vec a.\vec b + \vec b.\vec a + \vec b.\vec b\]
\[ \Rightarrow {\left| {\vec c} \right|^2} = \vec a.\vec a + \vec a.\vec b + \vec b.\vec a + \vec b.\vec b\]
\[ \Rightarrow {\left| {\vec c} \right|^2} = {\left| {\vec a} \right|^2} + 2\vec a.\vec b + {\left| {\vec b} \right|^2}\]-----(6)
Since \[\left| {\vec a} \right| = a\], \[\left| {\vec b} \right| = b\], \[\left| {\vec c} \right| = c\] then the equation (6) becomes
\[{c^2} = {a^2} + 2\vec a.\vec b + {b^2}\]----(7)
From the equations (5) and (7), we get
\[{a^2} + {b^2} + 2ab = {a^2} + 2\vec a.\vec b + {b^2}\]--(8)
On simplifying the equation (8), we get
\[ab = \vec a.\vec b\]---(9)
Using the equation (9) in the equation (1), we get
\[\cos \theta = 1\]
⟹ \[\theta = 2n\pi \]for \[n = 0,1,2,...\]
So, the correct answer is “Option C”.

Note:
Note that physical quantities are classified into two types, namely scalar and vector quantities. A scalar quantity is the physical quantity having only magnitude and doesn't have direction. Some examples of scalar quantities are Mass, Length, Time, Electric current, Temperature, etc. The main difference between scalar and vector quantity is that vector has direction whereas scalar doesn’t have direction.