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 $\Theta$ 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}
$