If the distance between the planes $4x - 2y - 4z + 1 = 0$ and $4x - 2y - 4z + d = 0$ is $7$,then the value of $d$ is:

The equation of the plane perpendicular to the line $\frac{x}{1}=\frac{y}{2}=\frac{z}{3}$ and passing through the point $(2, 3, 4)$ is:

The unit vector perpendicular to the plane $4x - 3y + 12z = 15$ is

In a three-dimensional $xyz$ space,the equation $x^2 - 5x + 6 = 0$ represents:

If the cartesian equation of the plane passing through the point $\hat{i}+2 \hat{j}+\hat{k}$ and parallel to the vectors $2 \hat{i}+3 \hat{j}+\hat{k}$ and $-\hat{i}+2 \hat{j}-3 \hat{k}$ is $a x+b y+c z=1$,then $18(a+b+c)$ is equal to

Let $P$ be an arbitrary point such that the sum of the squares of the distances from the planes $x + y + z = 0$,$lx - nz = 0$,and $x - 2y + z = 0$ is equal to $9$. If the locus of the point $P$ is $x^2 + y^2 + z^2 = 9$,then the value of $l - n$ is equal to ......