Find the distance of the point $(0, 0, 0)$ from the plane $3x - 4y + 12z = 3$. (in $/13$)

The angle between the planes $\vec{r} \cdot(2 \hat{i}+4 \hat{j}-3 \hat{k})=5$ and $\vec{r} \cdot(5 \hat{i}+3 \hat{j}+4 \hat{k})=7$ is

If $A(-1, 2, 3)$,$B(1, 1, 1)$ and $C(2, -1, 3)$ are points on a plane,then a unit normal vector to the plane $ABC$ is:

Let $R^3$ denote the three-dimensional space. Take two points $P=(1, 2, 3)$ and $Q=(4, 2, 7)$. Let $\operatorname{dist}(X, Y)$ denote the distance between two points $X$ and $Y$ in $R^3$. Let
$S=\{X \in R^3: (\operatorname{dist}(X, P))^2 - (\operatorname{dist}(X, Q))^2 = 50\}$
$T=\{Y \in R^3: (\operatorname{dist}(Y, Q))^2 - (\operatorname{dist}(Y, P))^2 = 50\}$
Then which of the following statements is (are) $TRUE$?
$(A)$ There is a triangle whose area is $1$ and all of whose vertices are from $S$.
$(B)$ There are two distinct points $L$ and $M$ in $T$ such that each point on the line segment $LM$ is also in $T$.
$(C)$ There are infinitely many rectangles of perimeter $48$,two of whose vertices are from $S$ and the other two vertices are from $T$.
$(D)$ There is a square of perimeter $48$,two of whose vertices are from $S$ and the other two vertices are from $T$.

The plane $ax + by + cz = 1$ meets the coordinate axes at points $A, B$,and $C$. Find the centroid of the triangle $ABC$.

The plane $2x + 3y + 4z = 1$ meets the $X$-axis at $A$,the $Y$-axis at $B$,and the $Z$-axis at $C$. Then the centroid of $\triangle ABC$ is: