Maharashtra Board Solutions for Chapter: Line and Plane, Exercise 4: Exercise 6.3

Reduce the equation $\overrightarrow{r}·\left(3\hat{i}+4\hat{j}+12\hat{k}\right)=78$ to normal form and hence find

(i) the length of the perpendicular from the origin to the plane (ii) direction cosines of the normal.

Find the vector equation of the plane passing through the point having position vector $\hat{i}+\hat{j}+\hat{k}$ and perpendicular to the vector $4\hat{i}+5\hat{j}+6\hat{k}$.

Find the Cartesian equation of the plane passing through $A\left(-1,2,3\right)$, the direction ratios of whose normal are $0,2,5$.

Find the Cartesian equation of the plane passing through $A\left(7,8,6\right)$ and parallel to the $XY$ plane.

The foot of the perpendicular drawn from the origin to a plane is $M\left(1,0,0\right)$. Find the vector equation of the plane.

Find the vector equation of the plane passing through the point $A\left(-2,7,5\right)$ and parallel to vectors $4\hat{i}-\hat{j}+3\hat{k}$ and $\hat{i}+\hat{j}+\hat{k}$.

Find the Cartesian equation of the plane $\overrightarrow{r}=\left(5\hat{i}-2\hat{j}-3\hat{k}\right)+\lambda \left(\hat{i}+\hat{j}+\hat{k}\right)+\mu \left(\hat{i}-2\hat{j}+3\hat{k}\right)$.

Find the vector equation of the plane which makes intercepts $1,1,1$ on the co-ordinates axes.