If \(\alpha\) is the angle between two vectors \(\mathbf{p}=3 \hat{\mathbf{i}}+4…
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 \(\sin (\alpha)=\)
\(\sqrt{\frac{145}{156}}\)
\(\sqrt{\frac{135}{156}}\)
\(\sqrt{\frac{155}{156}}\)
\(\sqrt{\frac{165}{156}}\)
Solution
If \(\alpha\) is an acute angle between vectors \(p\) and \(q\), then
\(\begin{aligned}
\operatorname{Sin} \alpha & =\frac{|\mathbf{p} \times \mathbf{q}|}{|\mathbf{p}||\mathbf{q}|}=\frac{\sqrt{(4-1)^2+(-3-2)^2+(-3-8)^2}}{\sqrt{9+16+1} \sqrt{4+1+1}} \\
& =\frac{\sqrt{9+25+121}}{\sqrt{156}}=\sqrt{\frac{155}{156}}
\end{aligned}\)