Angle Between Two PlanesAngle Between Two Planes - Planes & Angles - Practice Questions & MCQ

The angle between two planes is defined as the angle between their normals.
Let $\mathrm{\Theta }$ be the angle between two planes $\overrightarrow{\mathbf{r}}\cdot {\overrightarrow{\mathbf{n}}}_1={\mathbf{d}}_1$ and $\overrightarrow{\mathbf{r}}\cdot {\overrightarrow{\mathbf{n}}}_2={\mathbf{d}}_2$ then,
$\cos \theta =\frac{{\overrightarrow{\mathbf{n}}}_1\cdot {\overrightarrow{\mathbf{n}}}_2}{\left|{\overrightarrow{\mathbf{n}}}_1\right|\left|{\overrightarrow{\mathbf{n}}}_1\right|}$

Condition for Perpendicularity of Two Plane

If the planes $\overrightarrow{\mathbf{r}}\cdot {\overrightarrow{\mathbf{n}}}_1={\mathbf{d}}_1$ and $\overrightarrow{\mathbf{r}}\cdot {\overrightarrow{\mathbf{n}}}_2={\mathbf{d}}_2$ are perpendicular, then ${\overrightarrow{\mathbf{n}}}_1$ and ${\overrightarrow{\mathbf{n}}}_2$ are perpendicular.
Therefore ${\overrightarrow{\mathbf{n}}}_1\cdot {\overrightarrow{\mathbf{n}}}_2=0$.

Condition for Parallelism of Two Plane
If the planes $\overrightarrow{\mathbf{r}}\cdot {\overrightarrow{\mathbf{n}}}_1={\mathbf{d}}_1$ and $\overrightarrow{\mathbf{r}}\cdot {\overrightarrow{\mathbf{n}}}_2={\mathbf{d}}_2$ are parallel, then there exists a scalar $\lambda$ such that ${\overrightarrow{\mathbf{n}}}_1=\lambda {\overrightarrow{\mathbf{n}}}_2$.

Cartesian Form

Let $\theta$ be the angle between the planes, $a_{1}x+b_{1}y+c_{1}z+d_1=0$ and $a_{2}x+b_{2}y+c_{2}z+d_2=0$.

Then,

$\cos \theta =\left|\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}\right|$

The direction ratios of the normal to the planes are ${\mathrm{a}}_1,{\mathrm{b}}_1,{\mathrm{c}}_1$ and ${\mathrm{a}}_2,{\mathrm{b}}_2{\mathrm{c}}_2$ respectively.

Condition for Perpendicularity of Two Planes
If the planes are at right angles, then $\theta ={90}^{\circ }$ and so $\cos \theta =0$. Hence, $\cos \theta =a_1{a}_2+b_1{b}_2+c_1{c}_2=0$.

Condition for Parallelism of Two Plane
$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$