# LinAlg4.2.18

Reduce the equation $\bar{r}\cdot(4\bar{i}-12\bar{j}-3\bar{k})+7=0\,$ to normal form and hence find the length of the perpendicular from the origin to the plane.

Given $\bar{r}\cdot(4\bar{i}-12\bar{j}-3\bar{k})+7=0\,$

Simplifying,we have $\bar{r}\cdot(-4\bar{i}+12\bar{j}+3\bar{k})=7\,$

This is in the form of $\bar{r}\cdot\bar{n}=p\,$ where $\bar{n}=-4\bar{i}+12\bar{j}+3\bar{k}\,$

Since $|n|=\sqrt{(-4)^2+(12)^2+3^2}=13\,$ not equal to zero,the given equation is not in the normal form.We divide both sides by 13.

Therefore the equation of the plane in the normal form is $\bar{r}\cdot\frac{1}{13}(-4\bar{i}+12\bar{j}+3\bar{k})=\frac{7}{13}\,$

Therefore,the length of the perpendicular is $\frac{7}{13}\,$

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