Find the equation of the plane passing through the point $(1, 2, 3)$ and parallel to the plane $2x + 3y - 4z = 0$.

The Cartesian equation of the plane passing through the point $(1, -2, 3)$ and perpendicular to the vector $-\hat{i} + 2\hat{j} - 3\hat{k}$ is:

If $M$ is the foot of the perpendicular drawn from $P(1, 2, -1)$ to the plane passing through the point $A(3, -2, 1)$ and perpendicular to the vector $\vec{n} = 4\hat{i} + 7\hat{j} - 4\hat{k}$,then the length of $PM$,in proper units,is

In space,the equation $by + cz + d = 0$ represents a plane perpendicular to the plane:

The plane $ax + by = 0$ is rotated about its line of intersection with the plane $z = 0$ through an angle $\alpha$. Prove that the equation of the plane in its new position is $ax + by \pm (\sqrt{a^{2} + b^{2}} \tan \alpha) z = 0$.

Find the equation of the plane passing through the points $(2, 1, 0)$,$(3, -2, -2)$,and $(3, 1, 7)$.