Find the equation of the plane passing through the points $A(2, 2, -1)$,$B(3, 4, 2)$,and $C(7, 0, 6)$.

The direction ratios of the normal to the plane passing through $(1, 0, 0)$ and $(0, 1, 0)$ which makes an angle $\frac{\pi}{4}$ with the plane $x + y = 3$ are:

In the following cases,determine whether the given planes are parallel or perpendicular,and in case they are neither,find the angles between them: $2x - 2y + 4z + 5 = 0$ and $3x - 3y + 6z - 1 = 0$.

Let the plane $ax+by+cz+d=0$ bisect the line segment joining the points $P(4,-3,1)$ and $Q(2,3,-5)$ at right angles. If $a, b, c, d$ are integers,then the minimum value of $(a^{2}+b^{2}+c^{2}+d^{2})$ is

If $-2, \frac{4}{3}, \frac{-4}{5}$ are the intercepts made by a plane on $X, Y, Z$-axes respectively,then the direction cosines of a normal to this plane are

For what value of $k$ do the planes $kx + 4y + z = 0$,$4x + ky + 2z = 0$,and $2x + 2y + z = 0$ intersect in a single line?