Equation of a Plane in Normal Form

Find the equation of the plane passing through the point $(2, 4, 6)$ and making equal intercepts on the coordinate axes.

A plane meets the coordinate axes at $A,B$ and $C,$ respectively, such that the centroid of triangle $ABC$ is $(1, -2, 3).$ Find the equation of the plane.

Find the vector equation of a plane which is at a distance of $5$ units from the origin and its normal vector is $2\hat{i}-3\hat{j}+6\hat{k}$.

Write the equation of a plane which is at a distance of $5\sqrt{3}$ units from origin and the normal to which is equally inclined to coordinate axes.

Write the plane $\overrightarrow{r}·\left(2\hat{i}+3\hat{j}-6\hat{k}\right)=14$ in normal form.

Write a vector normal to the plane $\overrightarrow{r}=l\overrightarrow{b}+m\overrightarrow{c}$.

Find the vector and Cartesian forms of the equation of the plane passing through the point $(1, 2, -4)$ and parallel to the lines $\overrightarrow{r}=\left(\hat{i}+2\hat{j}-4\hat{k}\right)+\lambda \left(2\hat{i}+3\hat{j}+6\hat{k}\right)$ and $\overrightarrow{r}=\left(\hat{i}-3\hat{j}+5\hat{k}\right)+\mu \left(\hat{i}+\hat{j}-\hat{k}\right)$ Also, find the distance of the point $(9, -8, -10)$ from the plane thus obtained.

Find the equation of the plane that contains the point $\left(1,-1,2\right)$ and is perpendicular to each of the planes $2x+3y-2z=5$ and $x+2y-3z=8$.

Find the equation of the plane passing through the point $\left(-1,3,2\right)$ and perpendicular to each of the planes $x+2y+3z=5$ and $3x+3y+z=0$.

Find the Cartesian forms of the equations of the following planes.

$\overrightarrow{r}=\left(\hat{i}-\hat{j}\right)+s\left(-\hat{i}+\hat{j}+2\hat{k}\right)+t\left(\hat{i}+2\hat{j}+\hat{k}\right)$

Find the equation of the plane passing through the points whose coordinates are $\left(-1,1,1\right)$ and $\left(1,-1,1\right)$ and perpendicular to the plane $x+ 2y+2z=5$.

Find the Cartesian forms of the equations of the plane.

$\overrightarrow{r}=\left(1+s+t\right)\hat{i}+\left(2-s+t\right)\hat{i}+\left(3-2s+2t\right)\hat{k}$

Find the vector equation of the plane through the points $\left(2,1,-1\right)$ and $\left(-1,3,4\right)$ and perpendicular to the plane $x-2y+4z=10$.

Find the equation of the plane passing through the points $\left(2,2,1\right)$ and $\left(9,3,6\right)$ and perpendicular to the plane $2x+6y+6z=1$.

Find the equation of the plane passing through the origin and perpendicular to each of the planes $x+2y-z=1$ and $3x-4y+z=5$.

Find the equation of the plane passing through the points $\left(1,-1,2\right)$ and $\left(2,-2,2\right)$ and which is perpendicular to the plane $6x-2y+2z=9$.

Obtain the equation of the plane passing through the point $\left(1,-3,-2\right)$ and perpendicular to the planes $x+2y+2z=5$ and $3x+3y+2z=8$.

Find the vector equation of a plane which is at a distance of $5$ units from the origin and which is normal to the vector $\hat{i}-2\hat{j}-2\hat{k}$

Find the vector equation of the plane which is at a distance of $\frac{6}{\sqrt{29}}$ from the origin and its normal vector from the origin is $2\hat{i}-3\hat{j}+4\hat{k}$. Also, find its Cartesian form.

Find a unit normal vector to the plane $x+2y+3z-6=0$