The vector equation of the plane passing thro | vedclass

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(B) The equation of a plane parallel to the plane $\vec{r} \cdot (4\hat{i} - 12\hat{j} - 3\hat{k}) - 7 = 0$ is given by $\vec{r} \cdot (4\hat{i} - 12\

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$\vec{r} \cdot (4\hat{i} - 12\hat{j} - 3\hat{k}) = 32$

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(B) The equation of a plane parallel to the plane $\vec{r} \cdot (4\hat{i} - 12\hat{j} - 3\hat{k}) - 7 = 0$ is given by $\vec{r} \cdot (4\hat{i} - 12\hat{j} - 3\hat{k}) + \lambda = 0$.Since this plane passes through the point with position vector $\vec{a} = 2\hat{i} - \hat{j} - 4\hat{k}$,we substitute this into the equation:$(2\hat{i} - \hat{j} - 4\hat{k}) \cdot (4\hat{i} - 12\hat{j} - 3\hat{k}) + \lambda = 0$Calculating the dot product:$(2)(4) + (-1)(-12) + (-4)(-3) + \lambda = 0$$8 + 12 + 12 + \lambda = 0$$32 + \lambda = 0$$\lambda = -32$Substituting $\lambda = -32$ back into the equation,we get:$\vec{r} \cdot (4\hat{i} - 12\hat{j} - 3\hat{k}) - 32 = 0$$\vec{r} \cdot (4\hat{i} - 12\hat{j} - 3\hat{k}) = 32$Thus,the correct option is $B$.

The vector equation of the plane passing through the point $2\hat{i} - \hat{j} - 4\hat{k}$ and parallel to the plane $\vec{r} \cdot (4\hat{i} - 12\hat{j} - 3\hat{k}) - 7 = 0$ is:

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