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

$A$ plane passing through $(-1, 2, 3)$ and whose normal makes equal angles with the coordinate axes is

Let $\pi_1$ be the plane passing through the point $2\hat{i}-\hat{j}+\hat{k}$ and perpendicular to the vector $a\hat{i}+2\hat{j}-3\hat{k}$, and $\pi_2$ be the plane passing through the point $\hat{i}+2\hat{j}-\hat{k}$ and perpendicular to the vector $\hat{i}-2\hat{j}+\hat{k}$. If $\theta$ is the angle between the planes $\pi_1$ and $\pi_2$ and $\cos \theta = -\sqrt{\frac{3}{7}}$, then the integral value of $a$ is:

$A$ plane meets the coordinate axes at $A, B, C$ and $(\alpha, \beta, \gamma)$ is the centroid of the triangle $ABC$. Then the equation of the plane is

$A$ plane meets the coordinate axes at $P, Q,$ and $R$ such that the position vector of the centroid of $\Delta PQR$ is $2i - 5j + 8k$. Then the equation of the plane is:

Find the coordinates of the foot of the perpendicular drawn from the origin to the plane $5y + 8 = 0$.