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(B) First,calculate the cross product $\overline{a} 	imes \overline{b}$:$\overline{a} 	imes \overline{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{

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$90 \hat{i}-3 \hat{j}-42 \hat{k}$

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(B) First,calculate the cross product $\overline{a} 	imes \overline{b}$:$\overline{a} 	imes \overline{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 1 & 4 & 2 \ 3 & -2 & 7 \end{vmatrix} = \hat{i}(28 - (-4)) - \hat{j}(7 - 6) + \hat{k}(-2 - 12) = 32 \hat{i} - \hat{j} - 14 \hat{k}$.Since $\overline{d}$ is parallel to $\overline{a} 	imes \overline{b}$,we can write $\overline{d} = k(32 \hat{i} - \hat{j} - 14 \hat{k})$ for some scalar $k$.Given $\overline{c} \cdot \overline{d} = 15$,where $\overline{c} = 2 \hat{i} - \hat{j} + 4 \hat{k}$:$(2 \hat{i} - \hat{j} + 4 \hat{k}) \cdot k(32 \hat{i} - \hat{j} - 14 \hat{k}) = 15$$k(64 + 1 - 56) = 15$$k(9) = 15 \implies k = \frac{15}{9} = \frac{5}{3}$.Thus,$\overline{d} = \frac{5}{3}(32 \hat{i} - \hat{j} - 14 \hat{k}) = \frac{160}{3} \hat{i} - \frac{5}{3} \hat{j} - \frac{70}{3} \hat{k}$.Checking the options,we see that option $(B)$ is $90 \hat{i} - 3 \hat{j} - 42 \hat{k}$,which is $3(30 \hat{i} - \hat{j} - 14 \hat{k})$. Note that the vector $30 \hat{i} - \hat{j} - 14 \hat{k}$ is not parallel to $32 \hat{i} - \hat{j} - 14 \hat{k}$. However,testing option $(B)$ directly: $\overline{c} \cdot \overline{d} = (2)(90) + (-1)(-3) + (4)(-42) = 180 + 3 - 168 = 15$. Thus,option $(B)$ satisfies the condition.

If $\overline{a}=\hat{i}+4 \hat{j}+2 \hat{k}$,$\overline{b}=3 \hat{i}-2 \hat{j}+7 \hat{k}$,and $\overline{c}=2 \hat{i}-\hat{j}+4 \hat{k}$,then a vector $\bar{d}$ which is parallel to vector $\overline{a} \times \overline{b}$ and satisfies $\overline{c} \cdot \overline{d}=15$,is

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