Angle between Two Planes

Find the value of $p$, if the angle between the planes $\overrightarrow{r}·\left(p\hat{i}-\hat{j}+2\hat{k}\right)+3=0$ and $\overrightarrow{r}.(2\hat{i}-p\hat{j}-\hat{k})-5=0$ is $\frac{\pi }{3}$.

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

The plane $lx+my=0$ is rotated an angle $\mathrm{\alpha }$ about its line of intersection with the plane $z=0$ , then the equation to the plane in its new position is

The equation of the plane passing through the points $\left(1,-3,-2\right)$ and perpendicular to planes $x+2y+2z=5$ and $3x+3y+2z=8$ , is

In the space the equation $by+cz+d=0$ represents a plane perpendicular to the plane

The angle between two planes is equal to

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

The angle between the planes $3x-4y+5z=0$ and $2x-y-2z=5$ is

Direction ratios of two lines are $a,\mathrm{ }b,\mathrm{ }c$ and $\frac{1}{bc},\frac{1}{ca},\frac{1}{ab}$. The lines are

Find the measure of the angle between the planes $2x-y+z=2$ and $x+y+2z=3$

The equation of the plane passes through $\text{P}\left(2,\mathrm{ }3,\mathrm{ }4\right)$ and parallel to the plane $\mathrm{x}+2\mathrm{y}+4\mathrm{z}=5$ is -