Question

If ${\vec a_1}$ and ${\vec a_2}$ are two non collinear unit vector and if $\left| {{{\vec a}_1} + {{\vec a}_2}} \right| = \sqrt 3 $ then find the value of $({\vec a_1} - {\vec a_2})(2{\vec a_1} + {\vec a_2})$
(A) 2
(B) $\dfrac{3}{2}$
(C) $\dfrac{1}{2}$
(D) 1

Answer 1

Hint: The relation between the two unit vectors is given by $\left| {{{\vec a}_1} + {{\vec a}_2}} \right| = \sqrt 3 $. Squaring on both sides and simplifying the dot product find the value of $\theta $. Now, expand $({\vec a_1} - {\vec a_2})(2{\vec a_1} + {\vec a_2})$ and after simplifying the dot product again substitute the values of both vectors and $\theta $. On evaluating further, we get the required value.

Complete step-by-step solution:
${\vec a_1}$ and ${\vec a_2}$ are two non-collinear unit vectors so their magnitude is equal one.
It is given that, $\left| {{{\vec a}_1} + {{\vec a}_2}} \right| = \sqrt 3 $
Squaring on both sides.
${\left| {{{\vec a}_1} + {{\vec a}_2}} \right|^2} = {\left( {\sqrt 3 } \right)^2}$
${\left| {{{\vec a}_1}} \right|^2} + {\left| {{{\vec a}_2}} \right|^2} + 2{\vec a_1}.{\vec a_2} = 3$
$1 + 1 + 2\left| {{{\vec a}_1}} \right|\left| {{{\vec a}_2}} \right|\cos \theta = 3$
$2 \times 1 \times 1 \times \cos \theta = 3 - 2$
$\cos \theta = \dfrac{1}{2}$
We need to find the value of $({\vec a_1} - {\vec a_2})(2{\vec a_1} + {\vec a_2})$
Multiply both to get four individual terms.
 \[ = 2{\left| {{{\vec a}_1}} \right|^2} - 2{\vec a_1}.{\vec a_2} + {\vec a_1}.{\vec a_2} - {\left| {{{\vec a}_2}} \right|^2}\]
 $ = 2 - {\vec a_1}.{\vec a_2} - 1$
 $ = 1 - \left| {{{\vec a}_1}} \right|\left| {{{\vec a}_2}} \right|\cos \theta $
Substitute the value of $\theta $ .
 $ = 1 - 1 \times 1 \times \dfrac{1}{2}$
 $ = \dfrac{1}{2}$

Hence, the value of $({\vec a_1} - {\vec a_2})(2{\vec a_1} + {\vec a_2})$ is $\dfrac{1}{2}$ and the correct option is C.

Note: Collinear vectors are those vectors which act along the same line. So, the angle between them can be zero or ${180^0}$. Coplanar vectors are where three vectors lie in the same plane. Equal, parallel, anti-parallel, collinear, zero, unit, orthogonal, polar, axial, coplanar and negative are the types of vectors.