If $\alpha$ is the angle between two vectors $\mathbf{p}=3 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}-\hat{\mathbf{k}}$ and $\mathbf{q}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$, then $\sin (\alpha)=$

Question

If $\alpha$ is the angle between two vectors $\mathbf{p}=3 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}-\hat{\mathbf{k}}$ and $\mathbf{q}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$, then $\sin (\alpha)=$, $\sqrt{\frac{145}{156}}$, $\sqrt{\frac{135}{156}}$, $\sqrt{\frac{155}{156}}$, $\sqrt{\frac{165}{156}}$

MARKS App · Accepted Answer

If $\alpha$ is an acute angle between vectors $p$ and $q$, then $\begin{aligned} \operatorname{Sin} \alpha & =\frac{|\mathbf{p} \times \mathbf{q}|}{|\mathbf{p}||\mathbf{q}|}=\frac{\sqrt{(4-1)^2+(-3-2)^2+(-3-8)^2}}{\sqrt{9+16+1} \sqrt{4+1+1}} \ & =\frac{\sqrt{9+25+121}}{\sqrt{156}}=\sqrt{\frac{155}{156}} \end{aligned}$

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