The length of the perpendicular from the orig | vedclass

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(D) The length of the perpendicular from a point with position vector $\bar{a}$ to the plane $\bar{r} \cdot \bar{n} = p$ is given by the formula $d =

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$\sqrt{14}$ units

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(D) The length of the perpendicular from a point with position vector $\bar{a}$ to the plane $\bar{r} \cdot \bar{n} = p$ is given by the formula $d = \frac{|\bar{a} \cdot \bar{n} - p|}{|\bar{n}|}$.Here,the origin is the point,so $\bar{a} = 0\hat{i} + 0\hat{j} + 0\hat{k}$.The plane equation is $\bar{r} \cdot (\hat{i} - 2\hat{j} + 3\hat{k}) = 14$,so $\bar{n} = \hat{i} - 2\hat{j} + 3\hat{k}$ and $p = 14$.Calculating the dot product: $\bar{a} \cdot \bar{n} = 0(1) + 0(-2) + 0(3) = 0$.Calculating the magnitude of the normal vector: $|\bar{n}| = \sqrt{1^2 + (-2)^2 + 3^2} = \sqrt{1 + 4 + 9} = \sqrt{14}$.Substituting these values into the formula: $d = \frac{|0 - 14|}{\sqrt{14}} = \frac{14}{\sqrt{14}} = \sqrt{14}$ units.

The length of the perpendicular from the origin to the plane $\bar{r} \cdot (\hat{i} - 2\hat{j} + 3\hat{k}) = 14$ is

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