Question

Let $\vec{a}$ and $\vec{b}$ be two non-zero vectors perpendicular to each other and $|\vec{a}|=|\vec{b}| .$ If $|\vec{ a } \times \vec{ b }|=|\vec{ a }|$, then the angle between the vectors $(\vec{a}+\vec{b}+(\vec{a} \times \vec{b}))$ and $\vec{a}$ is equal to :

• A. $\sin ^{-1}\left(\frac{1}{\sqrt{3}}\right)$
• B. $\cos ^{-1}\left(\frac{1}{\sqrt{3}}\right)$
• C. $\cos ^{-1}\left(\frac{1}{\sqrt{2}}\right)$
• D. $\sin ^{-1}\left(\frac{1}{\sqrt{6}}\right)$

Answer 1

Answer: (B)

$|\vec{ a }|=|\vec{ b }|,|\vec{ a } \times \vec{ b }|=|\vec{ a }|, \vec{ a } \perp \vec{ b }$
$|\vec{a} \times \vec{b}|=|\vec{a}| \Rightarrow |\vec{a}||\vec{b}| \sin 90^{\circ}=|\vec{a}| \Rightarrow |\vec{b}|=1=|\vec{a}|$
$\vec{a}$ and $\vec{b}$ are mutually perpendicular unit
vectors.
Let $\vec{ a }=\hat{ i }, \vec{ b }=\hat{ j } \Rightarrow \vec{ a } \times \vec{ b }=\hat{ k }$
$\cos \theta=\frac{(\hat{ i }+\hat{ j }+\hat{ k }) \hat{ i }}{\sqrt{3} \sqrt{1}}=\frac{1}{\sqrt{3}} \Rightarrow \theta=\cos ^{-1}\left(\frac{1}{\sqrt{3}}\right)$