Planes in 3D

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 $\overrightarrow{c}=-8\widehat{\mathrm{i}}-\widehat{\mathrm{j}}+\mathrm{\lambda }\widehat{\mathrm{k}}$

If the plane $2x – 3y – 6z = 13$ makes an angle ${\sin }^{-1}\left(\lambda \right)$ with the $x -$ axis, then the value of $\lambda$ is 

The angle between the line $\overrightarrow{r}=(\hat{i}+2\hat{ȷ}-\hat{k})+\lambda (\hat{i}-\hat{j}+\hat{k})$ and the plane $\overrightarrow{r}·(2\hat{i}-\hat{ȷ}+\hat{k})=5$ is 

The angle between the plane $\overrightarrow{r}.\left(\hat{i}-2\hat{j}+3\hat{k}\right)=5$ and the line $\overrightarrow{r}=\left(\hat{i}+\hat{j}-\hat{k}\right)+\lambda \left(\hat{i}-\hat{j}+\hat{k}\right)$ is 

The angle between the line $\frac{x-1}{3}=\frac{y+1}{2}=\frac{z+2}{4}$ and the plane $2x+y-3z+4=0$ is 

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

Find the angle between the line $\overrightarrow{r}=(\hat{i}+2\hat{ȷ}-k)+\lambda (\hat{i}-\hat{j}+\hat{k})$ and the plane $\overrightarrow{r}\left(2\stackrel{⏜}{i}-\stackrel{⏜}{j}+\stackrel{⏜}{k}\right)=5$.

Reduce the equation ${\overrightarrow{r}}_1·(3i-4\hat{j}+12\hat{k})=3$ to normal form and  hence, find the length g perpendicular from the origin to the plane.

Test whether the lines $\overrightarrow{r}=\left(\hat{i}+\hat{j}-\hat{k}\right)+\lambda \left(3\hat{i}-\hat{j}\right)$ and $\overrightarrow{r}=\left(4\hat{i}-\hat{j}\right)+\mu \left(2\hat{i}+3\hat{k}\right)$ are coplanar. If so, find the equation of the plane containing these two lines

Find the angle between the line $\frac{x-1}{3}=\frac{y+1}{2}=\frac{z+2}{4}$ and the plane $2x+y-3z+4=0$

Find the angle between the planes $\overrightarrow{r}.\left(\hat{i}-2\hat{j}+3\hat{k}\right)=5$ and the line $\overrightarrow{r}=\left(\hat{i}+\hat{j}-\hat{k}\right)+\lambda \left(\hat{i}-\hat{j}+\hat{k}\right)$

The vector equation of the plane passing through the points $R(2,5,-3), S(-2,-3,5)$ and $T(5,3,-3)$ is

The equation of the plane passing through the points $\left(2, 3, 1\right), \left(4, -5, 3\right)$ and parallel to $y$-axis is

The angle between the line $\overrightarrow{r}=\left(i+\hat{\mathit{j}}-\hat{\mathit{k}}\right)+\lambda \left(3\hat{i}+\hat{\mathit{j}}\right)$ and the plane $\overrightarrow{r}·\left(\hat{\mathit{ı}}+2\hat{\mathit{ȷ}}+3\hat{\mathit{k}}\right)=8$ is

A plane passes through $\left(1,-2,1\right)$ and is perpendicular to two planes $2x-2y+z=0$ and  $x-y+2z=4.$ The distance of the plane from the point $\left(0,\mathrm{}2,\mathrm{}2\right)$ 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 $a^2+b^2+c^2+2abc$.

If $ax+by+cz=9$ is the equation of the plane through the points $(1, 1, 0)$ and $(2, 3, 2)$ & parallel to the line $\frac{x-1}{1}=\frac{y-1}{-2}=\frac{z-2}{3}$ then find $a+b+c$.

Show that the line of intersection of the planes $\overrightarrow{r}·(\hat{i}+3\hat{j}-2\hat{k})=0$ and $\overrightarrow{r}\left(2\hat{i}+4\hat{j}-3\hat{k}\right)=0$ is equally inclined to $\hat{i}$ & $\hat{j}$. Also find the angle it makes with $\hat{k}$.