The perpendicular distance from the origin to the plane containing the points having position vectors $\hat{i}+2\hat{j}+3\hat{k}$, $2\hat{i}+3\hat{j}-4\hat{k}$, and $3\hat{i}-4\hat{j}+5\hat{k}$ is

If the equation of the plane passing through the point $A(-2, 1, 3)$ and perpendicular to the vector $3 \hat{i} + \hat{j} + 5 \hat{k}$ is $ax + by + cz + d = 0$,then $\frac{a + b}{c + d} = $

$A$ plane passing through a point $(2,2,2)$ cuts the positive semi-axes at $A$,$B$,and $C$. If $P(\alpha, \beta, \gamma)$ is the centroid of the tetrahedron $OABC$ (where $O$ is the origin),then select the correct option.

If the line drawn from the point $(-2, -1, -3)$ meets a plane at a right angle at the point $(1, -3, 3)$,find the equation of the plane.

Let two planes be $P_1 : 2x - y + z = 2$ and $P_2 : x + 2y - z = 3$. Based on the given information,the equation of the acute angle bisector of the planes $P_1$ and $P_2$ 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: