Question

If the projection of the vector $\vec {a}$ on $\vec{b}$ is $\overrightarrow{a}$ on $\overrightarrow{b}$ is $|\overrightarrow{a}\times \overrightarrow{b}|$ and if $3\overrightarrow{b}=\vec{i}+\vec{j}+\vec{k},$ then the angle between $\vec{a}$ and $\vec{b}$ is

• A. $\pi /3$
• B. $\pi /2$
• C. $\pi /4$
• D. $\pi /6$

Answer 1

Answer: (A)

$\pi /3$

Answer 2

Given, projection of $\vec{a}$ on $\overrightarrow{b}=|\overrightarrow{a}\times \overrightarrow{b}|$
$\Rightarrow$ $\frac{\overrightarrow{a}\,.\,\overrightarrow{b}}{|\overrightarrow{b}|}=|\overrightarrow{a}\times \overrightarrow{b}|$
$\Rightarrow$ $\frac{|\overrightarrow{a}||\overrightarrow{b}|\cos \theta }{|\overrightarrow{b}|}=|\overrightarrow{a}||\overrightarrow{b}|\sin \theta$
$\Rightarrow$ $\tan \theta =\frac{1}{|\overrightarrow{b}|}$
$\Rightarrow$ $\tan \theta =\frac{1}{\frac{1}{3}\sqrt{{{1}^{2}}+{{1}^{2}}+{{1}^{2}}}}$
$\Rightarrow$ $\tan \theta =\sqrt{3}$
$\Rightarrow$ $\theta =\frac{\pi }{3}$