Question
If $$\vec a$$ and $$\vec b$$ are two unit vectors such that $$\vec a + 2\vec b$$ and $$5\vec a - 4\vec b$$ are perpendicular to each other then the angle between $$\vec a$$ and $$\vec b$$ is :
A.
$${45^ \circ }$$
B.
$${60^ \circ }$$
C.
$${\cos ^{ - 1}}\left( {\frac{1}{3}} \right)$$
D.
$${\cos ^{ - 1}}\left( {\frac{2}{7}} \right)$$
Answer :
$${60^ \circ }$$
Solution :
Given that $${\vec a}$$ and $${\vec b}$$ are two unit vectors
$$\therefore \left| {\vec a} \right| = 1{\text{ and }}\left| {\vec b} \right| = 1$$
Also, given that ($$\left( {\vec a + 2\vec b} \right) \bot \left( {5\vec a - 4\vec b} \right)$$
$$\eqalign{
& \Rightarrow \left( {\vec a + 2\vec b} \right).\left( {5\vec a - 4\vec b} \right) = 0 \cr
& \Rightarrow 5{\left| {\vec a} \right|^2} - 8{\left| {\vec b} \right|^2} - 4\vec a.\vec b + 10\vec b.\vec a = 0 \cr
& \Rightarrow 5 - 8 + 6\vec a.\vec b = 0 \cr
& \Rightarrow 6\left| {\vec a} \right|\,\left| {\vec b} \right|\,\cos \,\theta = 3 \cr} $$
[where $$\theta $$ is the angle between $${\vec a}$$ and $${\vec b}$$ ]
$$ \Rightarrow \cos \,\theta = \frac{1}{2}\,\,\,\,\, \Rightarrow \theta = {60^ \circ }$$