Find the vector and Cartesian equations of the plane that passes through the point $(1, 0, -2)$ and the normal to the plane is $\hat{i} + \hat{j} - \hat{k}$.

$A$ vector $\overrightarrow{V}$ in the first octant is inclined to the $x$-axis at $60^{\circ}$,to the $y$-axis at $45^{\circ}$ and to the $z$-axis at an acute angle. If a plane passing through the points $(\sqrt{2}, -1, 1)$ and $(a, b, c)$ is normal to $\overrightarrow{V}$,then:

The equation of the plane in normal form passing through the point $A(\vec{a})$, parallel to a vector $\vec{b}$ and containing a vector $\vec{c}$ is

Let two planes be $P_1: 2x - y + z = 2$ and $P_2: x + 2y - z = 3$. Find the equation of the plane passing through the point $(-1, 3, 2)$ and perpendicular to both planes $P_1$ and $P_2$.

If $(2, -3, 6)$ is the foot of the perpendicular drawn from the origin to a plane,then the equation of that plane is

Find the equation of the plane with intercepts $2, 3$ and $4$ on the $x, y$ and $z$-axis respectively.