Find the coordinates of the foot of the perpendicular drawn from the origin to the plane $3y + 4z - 6 = 0$.

The plane $\ell x+my=0$ is rotated about its line of intersection with the plane $z=0$ through an angle $\alpha$. The equation of the new plane is

The equation of a plane which cuts equal intercepts of unit length on the axes is:

The vector equation of the plane passing through the origin and the line of intersection of the planes $r \cdot a = \lambda$ and $r \cdot b = \mu$ is

Find the direction ratios of the normal to the plane passing through the points $(0, -1, 0)$ and $(0, 0, 1)$ and making an angle $\frac{\pi}{4}$ with the plane $y - z + 5 = 0$.

$A$ variable plane passes through a fixed point $(\alpha, \beta, \gamma)$ and meets the coordinate axes in $A, B$ and $C$. Let $P_1, P_2$ and $P_3$ be the planes passing through $A, B, C$ and parallel to the coordinate planes $YZ, ZX, XY$ respectively. Then,the locus of the point of intersection of the planes $P_1, P_2$ and $P_3$ is