Question

If the position vectors of P and Q are $\hat{i}+3\hat{j}-7\hat{k}\mathrm{and}5\hat{i}-2\hat{j}+4\hat{k}$ then the cosine of the angle between $\overrightarrow{PQ}$ and y-axis is
(a) $\frac{5}{\sqrt{162}}$

(b) $\frac{4}{\sqrt{162}}$

(c) $-\frac{5}{\sqrt{162}}$

(d) $\frac{11}{\sqrt{162}}$

Answer 1

(c) $-\frac{5}{\sqrt{162}}$

$$\begin{gathered} \overrightarrow{PQ}=\overrightarrow{OQ}-\overrightarrow{OP}=5\hat{i}-2\hat{j}+4\hat{k}-\left(\hat{i}+3\hat{j}-7\hat{k}\right)=4\hat{i}-5\hat{j}+11\hat{k} \\ \text{The unit vector along}\mathit{\text{y-}}\text{axis is}\hat{\mathrm{j}}\text{.} \\ \mathrm{Let}\mathrm{\theta }\mathrm{be}\mathrm{the}\mathrm{required}\mathrm{angle}. \\ \cos \mathrm{\theta }=\frac{\overrightarrow{PQ}.\hat{j}}{\left|\overrightarrow{PQ}\right|\left|\hat{j}\right|}=\frac{-5}{\sqrt{16+25+121}\sqrt{1}}=\frac{-5}{\sqrt{162}} \end{gathered}$$