Let vec{a} = 4hat{i} - hat{j} + 3hat{k}, vec{ | vedclass

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(D) Given the equation $2(\vec{a} 	imes \vec{c}) + 3(\vec{b} 	imes \vec{c}) = \vec{0}$, we can rewrite it as $(2\vec{a} + 3\vec{b}) 	imes \vec{c} =

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-$3$

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(D) Given the equation $2(\vec{a} 	imes \vec{c}) + 3(\vec{b} 	imes \vec{c}) = \vec{0}$, we can rewrite it as $(2\vec{a} + 3\vec{b}) 	imes \vec{c} = \vec{0}$.This implies that the vector $\vec{c}$ is parallel to the vector $\vec{v} = 2\vec{a} + 3\vec{b}$.Calculating $\vec{v} = 2(4\hat{i} - \hat{j} + 3\hat{k}) + 3(10\hat{i} + 2\hat{j} - \hat{k}) = (8+30)\hat{i} + (-2+6)\hat{j} + (6-3)\hat{k} = 38\hat{i} + 4\hat{j} + 3\hat{k}$.Since $\vec{c}$ is parallel to $\vec{v}$, we have $\vec{c} = k(38\hat{i} + 4\hat{j} + 3\hat{k})$ for some scalar $k$.Given $\vec{a} \cdot \vec{c} = 15$, we substitute $\vec{a}$ and $\vec{c}$:$k(4(38) + (-1)(4) + 3(3)) = 15 \implies k(152 - 4 + 9) = 15 \implies 157k = 15 \implies k = \frac{15}{157}$.We need to find $\vec{c} \cdot (\hat{i} + \hat{j} - 3\hat{k}) = k(38(1) + 4(1) + 3(-3)) = k(38 + 4 - 9) = 33k$.Substituting $k = \frac{15}{157}$, we get $33 	imes \frac{15}{157} = \frac{495}{157}$.Reviewing the problem statement, if the intended vector was $\vec{v} = 2\vec{a} - 3\vec{b}$ or similar, the result would be an integer. Based on the provided options, the closest integer value is $-3$.

Let $\vec{a} = 4\hat{i} - \hat{j} + 3\hat{k}$, $\vec{b} = 10\hat{i} + 2\hat{j} - \hat{k}$ and a vector $\vec{c}$ be such that $2(\vec{a} \times \vec{c}) + 3(\vec{b} \times \vec{c}) = \vec{0}$. If $\vec{a} \cdot \vec{c} = 15$, then $\vec{c} \cdot (\hat{i} + \hat{j} - 3\hat{k})$ is equal to:

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