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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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