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If the position vectors of $P$ and $Q$ are $\hat{i}+2 \hat{j}-7 \hat{k}$ and $5 \hat{i}-3 \hat{j}+4 \hat{k}$ respectively, then the cosine of the angle between $\overrightarrow{\mathrm{PQ}}$ and $\mathrm{z}$-axis is
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Verified Answer
The correct answer is:
$\frac{11}{\sqrt{162}}$
$\overrightarrow{P Q}=(5 \hat{i}-3 \hat{j}+4 \hat{k})-(\hat{i}+2 \hat{j}-7 \hat{k})$
$\overrightarrow{P Q}=4 \hat{i}-5 \hat{j}+11 \hat{k}$
Equation of $z$-axis : $\hat{k}$
$\begin{aligned}
\therefore \quad & \overrightarrow{P Q} \cdot \hat{k}=|\overrightarrow{P Q}| \cdot 1 \cdot \cos \theta \\
& \cos \theta=\frac{11}{\sqrt{4^2+5^2+11^2}}=\frac{11}{\sqrt{162}} .
\end{aligned}$
$\overrightarrow{P Q}=4 \hat{i}-5 \hat{j}+11 \hat{k}$
Equation of $z$-axis : $\hat{k}$
$\begin{aligned}
\therefore \quad & \overrightarrow{P Q} \cdot \hat{k}=|\overrightarrow{P Q}| \cdot 1 \cdot \cos \theta \\
& \cos \theta=\frac{11}{\sqrt{4^2+5^2+11^2}}=\frac{11}{\sqrt{162}} .
\end{aligned}$
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