Question

Find the angle which the vector $\overrightarrow{A}=3\hat{i}-2\hat{j}+4\hat{k}$ makes with $z-$axis.

• A. ${\cos }^{-1}\left(\frac{2}{\sqrt{29}}\right)$
• B. ${\cos }^{-1}\left(\frac{4}{\sqrt{29}}\right)$
• C. ${\cos }^{-1}\left(\frac{4}{\sqrt{9}}\right)$
• D. $0^{\circ }$

Answer 1

The correct option is B ${\cos }^{-1}\left(\frac{4}{\sqrt{29}}\right)$

Given, $\overrightarrow{A}=3\hat{i}-2\hat{j}+4\hat{k}$ and
other vector will be $\overrightarrow{B}=\hat{k}$, as it is $z-$axis
We know that,
$\cos \theta =\frac{\overrightarrow{A}.\overrightarrow{B}}{|\overrightarrow{A}||\overrightarrow{B}|}$
$\Rightarrow \cos \theta =\frac{(3\hat{i}-2\hat{j}+4\hat{k}).\left(\hat{k}\right)}{\sqrt{3^2+(-2{)}^2+4^2}\times \sqrt{1^2}}=\frac{4}{\sqrt{29}}$
$\Rightarrow \theta ={\cos }^{-1}\left(\frac{4}{\sqrt{29}}\right)$