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(C) The angle $\alpha$ between two vectors $p$ and $q$ satisfies $\sin(\alpha) = \frac{|p 	imes q|}{|p||q|}$.First,calculate the cross product $p 	i

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$\sqrt{\frac{155}{156}}$

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(C) The angle $\alpha$ between two vectors $p$ and $q$ satisfies $\sin(\alpha) = \frac{|p 	imes q|}{|p||q|}$.First,calculate the cross product $p 	imes q$:$p 	imes q = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 3 & 4 & -1 \ 2 & -1 & 1 \end{vmatrix} = \hat{i}(4 - 1) - \hat{j}(3 + 2) + \hat{k}(-3 - 8) = 3\hat{i} - 5\hat{j} - 11\hat{k}$.Next,calculate the magnitudes:$|p 	imes q| = \sqrt{3^2 + (-5)^2 + (-11)^2} = \sqrt{9 + 25 + 121} = \sqrt{155}$.$|p| = \sqrt{3^2 + 4^2 + (-1)^2} = \sqrt{9 + 16 + 1} = \sqrt{26}$.$|q| = \sqrt{2^2 + (-1)^2 + 1^2} = \sqrt{4 + 1 + 1} = \sqrt{6}$.Thus,$\sin(\alpha) = \frac{\sqrt{155}}{\sqrt{26} 	imes \sqrt{6}} = \frac{\sqrt{155}}{\sqrt{156}} = \sqrt{\frac{155}{156}}$.

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

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