Family of Planes

What would be the value of  $\lambda$  which makes the vectors  $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$  coplanar where  $\overrightarrow{a}=-4\widehat{\mathrm{i}}-6\widehat{\mathrm{j}}-2\widehat{\mathrm{k}},b=-\widehat{\mathrm{i}}+4\widehat{\mathrm{j}}+3\widehat{\mathrm{k}}$ and $\overrightarrow{c}=-8\widehat{\mathrm{i}}-\widehat{\mathrm{j}}+\mathrm{\lambda }\widehat{\mathrm{k}}$

Equation of the plane passing through the intersection of the planes $\overrightarrow{r}\cdot \left(2\widehat{\mathrm{i}}+\widehat{\mathrm{j}}+3\widehat{\mathrm{k}}\right)=7 \& \overrightarrow{r}\cdot \left(2\widehat{\mathrm{i}}+5\widehat{\mathrm{j}}+3\widehat{\mathrm{k}}\right)=9$ and the point $\left(2,1,3\right)$, is

Find the equation of plane passing through $x-$axis and points $\left(2,3,6\right)$ using family of planes passing through $x-$axis concept.

Find the equation of plane passing through $x-$axis and points $\left(2,2,4\right)$ using family of planes passing through $x-$axis concept.

Find the equation of plane passing through $x-$axis and points $\left(2,2,1\right)$ using family of planes passing through $x-$axis concept.

Find the equation of plane passing through $x-$axis and points $\left(3,2,3\right)$ using family of planes passing through $x-$axis concept.

Find the equation of plane passing through $x-$axis and points $\left(1,2,3\right)$ using family of planes passing through $x-$axis concept.

Find the axis to which the plane $3x+2y-4=0$ is parallel to. Find the distance from $\left(1,2,3\right)$ to plane.

Find the axis to which the plane $2z+x-2=0$ is parallel to. Find the distance from $\left(1,2,3\right)$ to plane.

Find the axis to which the plane $4x+2z=1$ is parallel to. Find the distance from $\left(1,2,3\right)$ to plane.

Find the axis to which the plane $2x+3z=4$ is parallel to. Find the distance from $\left(1,2,3\right)$ to plane.

The equation of the plane passing through the line of intersection of the planes $\overrightarrow{r}·\left(2\hat{i}-3\hat{j}+4\hat{k}\right)=1$ and $\overrightarrow{r}·\left(\hat{i}-\hat{j}\right)+4=0$ and perpendicular to the plane $\overrightarrow{r}·\left(2\hat{i}-\hat{j}-\hat{k}\right)+4=0$, is

Let $P$ be the plane, which contains the line of intersection of the planes, $x+y+z-6=0$ and $2x+3y+z+5=0$ and it is perpendicular to the $xy$-plane. Then the distance of the point $\left(0,0,256\right)$ from $P$ is equal to:

Let $P$ be the plane, which contains the line of intersection of the planes, $x+y+z-6=0$ and $2x+3y+z+5=0$ and it is perpendicular to the $xy$-plane. Then the distance of the point $\left(0,0,256\right)$ from $P$ is equal to:

The equation of the plane through the line of intersection of planes $2x+3y+4z-7=0$ and $x+y+z-1=0$ and perpendicular to the plane $x–5y+3z–2=0,$ is

Consider the planes $P_1:2x-y+2z=1,P_2:x+2y-z=2$ and $P_3:3x+6y-3z=6$. Then the number of point(s) where plane $P_1,P_2$ and $P_3$ intersect is/are

The equation of plane passing through the line of intersection of the planes $x+2y+3z=2$ and $x–y+z=3$ and perpendicular the plane $x–y+z–3=0$ is:

Let $\mathrm{P}$ be the point $(1, 2, 3)$ and $\mathrm{Q}$ be a point on the line $\overrightarrow{\mathrm{r}}=(\hat{\mathrm{i}}-\hat{\mathrm{j}}+5\hat{\mathrm{k}})+\lambda (-2\hat{\mathrm{i}}+3\hat{\mathrm{j}}+4\hat{\mathrm{k}}).$ Then the value of $\lambda$ for which line $\mathrm{PQ}$ is perpendicular to the plane $4x+9y-18z=1$ is-

If the planes $x-cy-bz=0, cx-y+az=0$ and $bx + ay-z=0$ pass through a straight line, then find the value of $a^2+b^2+c^2+2abc$.