Practicing Success
Let $\vec{a}, \vec{b}, \vec{c}$ be unit vectors such that $\vec{a} + \vec{b} + \vec{c} = \vec{x}, \vec{a} . \vec{x}=1, \vec{b} . \vec{x}=\frac{3}{2},|\vec{x}|=2$ Then angle between $\vec{c}$ and $\vec{x}$ is : |
$\cos ^{-1}\left(\frac{1}{4}\right)$ $\cos ^{-1}\left(\frac{3}{4}\right)$ $\cos ^{-1}\left(\frac{3}{8}\right)$ $\cos ^{-1}\left(\frac{5}{8}\right)$ |
$\cos ^{-1}\left(\frac{3}{4}\right)$ |
$\vec{a}+\vec{b}+\vec{c}=\vec{x}$ Taking dot with $\vec{x}$ on both sides, we get $\vec{x} . \vec{a}+\vec{x} . \vec{b}+\vec{x} . \vec{c}+\vec{x} . \vec{x}=|\vec{x}|^2=4$ $\Rightarrow 1+\frac{3}{2}+\vec{x} . \vec{c}=4$ $\Rightarrow \vec{x} . \vec{c}=\frac{3}{2}$ If '$\theta$' be the angle between $\vec{c}$ and $\vec{x}$ then $|\vec{x}||\vec{c}| \cos \theta=\frac{3}{2}$ $\Rightarrow \cos \theta=\frac{3}{4}$ $\Rightarrow \theta=\cos ^{-1}\left(\frac{3}{4}\right)$ Hence (2) is correct answer. |