If a = 2 hat{i} + 3 hat{j} - hat{k},b = hat{i | vedclass

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(D) vector that lies in the plane of $b$ and $c$ is given by a linear combination of $b$ and $c$. $A$ vector perpendicular to $a$ and lying in the pla

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$-17 \hat{i} - 21 \hat{j} - 97 \hat{k}$

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(D) vector that lies in the plane of $b$ and $c$ is given by a linear combination of $b$ and $c$. $A$ vector perpendicular to $a$ and lying in the plane of $b$ and $c$ is given by the vector product $a 	imes (b 	imes c)$.First,we calculate $b 	imes c$:$b 	imes c = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 1 & 2 & -5 \ 3 & 5 & -1 \end{vmatrix} = \hat{i}(-2 + 25) - \hat{j}(-1 + 15) + \hat{k}(5 - 6) = 23 \hat{i} - 14 \hat{j} - \hat{k}$.Now,we calculate $a 	imes (b 	imes c)$:$a 	imes (b 	imes c) = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 2 & 3 & -1 \ 23 & -14 & -1 \end{vmatrix} = \hat{i}(-3 - 14) - \hat{j}(-2 + 23) + \hat{k}(-28 - 69) = -17 \hat{i} - 21 \hat{j} - 97 \hat{k}$.Thus,the required vector is $-17 \hat{i} - 21 \hat{j} - 97 \hat{k}$.

If $a = 2 \hat{i} + 3 \hat{j} - \hat{k}$,$b = \hat{i} + 2 \hat{j} - 5 \hat{k}$,and $c = 3 \hat{i} + 5 \hat{j} - \hat{k}$,then a vector perpendicular to $a$ and in the plane containing $b$ and $c$ is:

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