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(C) The equation of the plane is given in the form $\bar{r} \cdot \bar{n} = d$,where $\bar{n} = 3 \hat{i} - 4 \hat{j} + 12 \hat{k}$ and $d = 8$. The l

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$\frac{8}{13}$ units

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(C) The equation of the plane is given in the form $\bar{r} \cdot \bar{n} = d$,where $\bar{n} = 3 \hat{i} - 4 \hat{j} + 12 \hat{k}$ and $d = 8$. The length of the perpendicular from the origin to the plane $\bar{r} \cdot \bar{n} = d$ is given by the formula $p = \frac{|d|}{|\bar{n}|}$. Here,$|\bar{n}| = \sqrt{3^2 + (-4)^2 + 12^2} = \sqrt{9 + 16 + 144} = \sqrt{169} = 13$. Therefore,the length of the perpendicular is $p = \frac{8}{13}$ units.

The length of the perpendicular from the origin to the plane $\bar{r} \cdot (3 \hat{i} - 4 \hat{j} + 12 \hat{k}) = 8$ is

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