$A$ point lying on the plane that passes through the points $\hat{i}-\hat{j}+\hat{k}$,$\hat{i}-2\hat{j}+3\hat{k}$,and $\hat{i}+2\hat{j}-3\hat{k}$ is:

If the position vectors of the points $A$ and $B$ are $3 \hat{i}+\hat{j}+2 \hat{k}$ and $\hat{i}-2 \hat{j}-4 \hat{k}$ respectively,then the equation of the plane passing through $B$ and perpendicular to $AB$ is

Let $P(\lambda, 2, 1)$ be a point on the plane which passes through the point $Q(4, -2, 2)$. If the plane is perpendicular to the line joining the points $A(-2, -21, 29)$ and $B(-1, -16, 33)$,then find the value of $\left(\frac{\lambda}{11}\right)^{2} - \frac{4\lambda}{11} - 4$.

$A$ plane $x$ passes through the point $(1, 1, 1)$. If $b, c, a$ are the direction ratios of a normal to the plane,where $a, b, c$ $(a < b < c)$ are the factors of $2001$,then the equation of the plane is

The equation of the plane passing through a point $A(2, -1, 3)$ and parallel to the vectors $\vec{a} = (3, 0, -1)$ and $\vec{b} = (-3, 2, 2)$ is:

$A$ plane passes through the point $A(2, 1, -3)$. If the distance of this plane from the origin is maximum,then its equation is