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  • Provide solution for RD Sharma maths class 12 chapter 23 Scaler and Dot Product exercise Multiple choice question 6

Provide solution for RD Sharma maths class 12 chapter 23 Scaler and Dot Product exercise Multiple choice question 6

Answers (1)

Answer:

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

Hint: You must know the formula of  \cos \theta=\frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}

Given:

\vec{P}=\hat{i}+3 \hat{j}-7 \hat{k} \text { and } \vec{Q}=5 \hat{i}-2 \hat{j}+4 \hat{k}

Solution:

Let O be the origin

\begin{aligned} &\overrightarrow{P Q}=\overrightarrow{O Q}-\overrightarrow{O P} \\\\ &\overrightarrow{P Q}=5 \hat{i}-2 \hat{j}+4 \hat{k}-(\hat{i}+3 \hat{j}-7 \hat{k}) \\\\ &\overrightarrow{P Q}=4 \hat{i}-5 \hat{j}+11 \hat{k} \end{aligned}

We know that unit vector along  y-\text { axis is } \hat{j}

Therefore, angle between \overrightarrow{P Q} \text { and } y-\text { axis }

\cos \theta=\frac{(\hat{4}-5 \hat{j}+11 \hat{k}) \hat{j}}{\sqrt{4^{2}+(-5)^{2}+11^{2}}}

\begin{aligned} &\cos \theta=\frac{0-5+0}{\sqrt{16+25+121}} \quad[\because \hat{i} \cdot \hat{j}=0, \hat{k} \cdot \hat{j}=0, \hat{j} \cdot \hat{j}=1] \\\\ &\cos \theta=\frac{-5}{\sqrt{162}} \end{aligned}

Hence, option (c) is correct

 

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