Let vec{r} cdot (3hat{i} - 2hat{j} + 7hat{k}) | vedclass

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(D) The normal vector to the plane $\vec{r} \cdot (3\hat{i} - 2\hat{j} + 7\hat{k}) = 32$ is $\vec{n} = 3\hat{i} - 2\hat{j} + 7\hat{k}$.The direction v

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$b = 18$

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(D) The normal vector to the plane $\vec{r} \cdot (3\hat{i} - 2\hat{j} + 7\hat{k}) = 32$ is $\vec{n} = 3\hat{i} - 2\hat{j} + 7\hat{k}$.The direction vector of the line is $\vec{v} = 5\hat{i} + b\hat{j} + 3\hat{k}$.Since the line is parallel to the plane, the line is perpendicular to the normal vector of the plane. Therefore, the dot product of $\vec{n}$ and $\vec{v}$ must be zero:$\vec{n} \cdot \vec{v} = 0$$(3)(5) + (-2)(b) + (7)(3) = 0$$15 - 2b + 21 = 0$$36 - 2b = 0$$2b = 36$$b = 18$.

Let $\vec{r} \cdot (3\hat{i} - 2\hat{j} + 7\hat{k}) = 32$ be the equation of a plane. If the line with direction ratios $(5, b, 3)$ is parallel to the plane, then the value of $b$ is:

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