Angle between Two Planes

If the equation of the plane passing through $(1,1,1)$ and parallel to the plane $x+2y+3z-7=0$ is of the form $x+2y+3z=k$ then find the value of $k$

If $\theta$ is the angle between the planes $x+2y+2z-5=0$ and $3x+3y+2z-8=0$ then $\theta =$

If $\theta$ is the acute angle between the planes $x+y-z=4 and x+2 y+z=9$ then, $\theta =$

The acute angle between the planes $2x-y+z=5\text{ and }x+y+2z=7$ is

The distance between the parallel planes $x+2y-2z+4=0$ and $x+2y-2z-8=0$ is 

The distance between the parallel planes $2x+3y+4z=4$ and $4x+6y+8z=12$ is 

Find the value of $\lambda$ for which the planes $x-4y+\lambda z+3=0\text{ and }2x+2y+3z=5$ are perpendicular to each other.

Find the equations of the planes parallel to the plane $x-2y+2z-4=0$, which are at a unit distance from the point $(1, 2, 3)$.

If the following planes are mutually perpendicular, then find the value of $\lambda$.

$$\begin{gathered} 2\mathrm{x}-4\mathrm{y}+3\mathrm{z}=5 \\ \mathrm{and} \\ \mathrm{x}+2\mathrm{y}+\mathrm{\lambda z}=5 \end{gathered}$$

 If the following planes are mutually perpendicular, then find the value of $\lambda$.

$\stackrel{\rightharpoonup }{\mathrm{r}}·\left(2\mathrm{i}-\mathrm{j}+\mathrm{\lambda k}\right)=5 \mathrm{and} \stackrel{\rightharpoonup }{\mathrm{r}}·\left(3\mathrm{i}+2\mathrm{j}+2\mathrm{k}\right)=4$

Prove that the following planes are mutually perpendicular: 

$\overrightarrow{r}·\left(2\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}\right)=4 \mathrm{and} \overrightarrow{r}·\left(-\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}\right)=3$

Find the angle between the given two planes.

$$\begin{gathered} \mathrm{x}+ \mathrm{y}-2\mathrm{z} = 3 \\ \mathrm{and} \\ 2\mathrm{x}-2\mathrm{y} + \mathrm{z} = 5 \end{gathered}$$

Find the angle between the two planes: 

$$\begin{gathered} 2\mathrm{x}-\mathrm{y} + \mathrm{z} = 4 \\ \mathrm{x}+\mathrm{y} + 2\mathrm{z} = 3 \end{gathered}$$

Find the angle between the planes: 

$\mathrm{x}+\mathrm{y}+2\mathrm{z}=9 \mathrm{and} 2\mathrm{x}-\mathrm{y}+\mathrm{z}=15$

Find the angle between the planes:

$\overrightarrow{r}·\left(\hat{i}+\hat{j}+2\hat{k}\right)=5$ and $\overrightarrow{r}·\left(2\hat{i}-\hat{j}+2\hat{k}\right)=6$

Find the angle between the planes: 

$\overrightarrow{r}·\left(2\hat{i}+3\hat{j}-6\hat{k}\right)=5$ and $\overrightarrow{r}·\left(\hat{i}-2\hat{j}+2\hat{k}\right)=9$

Find the angle between the planes:

$\overrightarrow{r}·\left(2\hat{i}-\hat{j}+2k\right)=6$ and $\overrightarrow{r}·\left(3\hat{i}+6\hat{j}-2\hat{k}\right)=9$

Value of $k$, for which the planes $x+2y+kz=0$ and $2x+y-2z=0$ are at right angles, is

If $\theta$ is the angle between the planes $2x-y+z=1$ and $x-2y+z+2=0$, then $\cos \theta =$

The angle between the planes $2x+y-2z=5$ and $3x-6y-2z=7$ is