The normal form of equation of a plane is \(\bar{r}.\hat{n} = p\) where \(\hat{n}\) is unit vector along the normal and p is the length of perpendicular drawn from origin to the plane.
Given plane is \(\bar{r}.(3\hat{i} + 4\hat{j} + 12\hat{k}) = 78 ....(1)\)
\(\bar{n}= 3\hat{i} + 4\hat{j} + 12\hat{k}\) is normal to the plane
Dividing both sides of (1) by 13, we get
This is the normal form of the equation of plane. Comparing with \(\bar{r}.\hat{n} = p\),
(i) the length of the perpendicular from the origin to plane is 6.
(ii) direction cosines of the normal are 3/13, 4/13, 12/13.
