If the image of the point $A(1, 1, 1)$ with respect to the plane $4x + 2y + 4z + 1 = 0$ is $B(\alpha, \beta, \gamma)$,then $\alpha + \beta + \gamma =$

$A$ plane at a unit distance from the origin intersects the three axes at points $P, Q,$ and $R$. If the centroid of $\Delta PQR$ is $(x, y, z)$,and it satisfies the equation $\frac{1}{x^2} + \frac{1}{y^2} + \frac{1}{z^2} = k$,then find the value of $k$.

Let $\alpha x+\beta y+\gamma z=1$ be the equation of a plane passing through the point $(3, -2, 5)$ and perpendicular to the line joining the points $(1, 2, 3)$ and $(-2, 3, 5)$. Then the value of $\alpha \beta \gamma$ is equal to $..........$.

$A$ 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.

Find the Cartesian equation of the following plane: $\vec{r} \cdot (2\hat{i} + 3\hat{j} - 4\hat{k}) = 1$

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