Angle Between Two Planes

Vector Form:

\[\cos\theta=\left|\frac{\overline{\mathbf{n₁}}.\overline{\mathbf{n₂}}}{\left|\overline{\mathbf{n₁}}\right|.\left|\overline{\mathbf{n₂}}\right|}\right|\]

Cartesian Form:

\[\cos\theta=\left|\frac{\mathrm{a}_{1}\mathrm{a}_{2}+\mathrm{b}_{1}\mathrm{b}_{2}+\mathrm{c}_{1}\mathrm{c}_{2}}{\sqrt{\mathrm{a}_{1}^{2}+\mathrm{b}_{1}^{2}+\mathrm{c}_{1}^{2}}\sqrt{\mathrm{a}_{2}^{2}+\mathrm{b}_{2}^{2}+\mathrm{c}_{2}^{2}}}\right|\]

Observe that if θ is an angle between the two planes, then so is 180 – θ Fig.

If `vec n _1` and `vec n_2` are normals to the planes and θ be the angle between the planes  
`vec r . vec n _1 = d_1` and `vec r . vec n _2 = d_2` . 
Then θ is the angle between the normals to the planes drawn from some common point We have  
cos θ = `|(vec n_1 . vec n_2)/ (|vec n _1| |vec n_2|)|`

Cartesian form 
Let θ be the angle between the planes, 
`A_1x + B_1y +C_1z + D_1 = 0` and `A_2x +B_2y + C_2 z + D_2 = 0`
The direction ratios of the normal to the planes are `A_1, B_1, C_1` and `A_2, B_2, C_2` respectively.
Therefore , cos θ = `|(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))|`

Vector Form:

\[\sin\theta=\left|\frac{\overline{\mathbf{b}}.\overline{\mathbf{n}}}{\left|\overline{\mathbf{b}}\right|.\left|\overline{\mathbf{n}}\right|}\right|\]

Cartesian Form:

\[\mathrm{sin}\theta=\frac{\mathrm{aa}_{1}+\mathrm{bb}_{1}+\mathrm{cc}_{1}}{\sqrt{\mathrm{a}^{2}+\mathrm{b}^{2}+\mathrm{c}^{2}}\sqrt{\mathrm{a}_{1}^{2}+\mathrm{b}_{1}^{2}+\mathrm{c}_{1}^{2}}}\]

Condition for Perpendicularity:

\[\overline{\mathbf{b}}=\lambda\overline{\mathbf{n}}\], λ is a parameter

\[\frac{\mathbf{a}_{1}}{\mathbf{a}}=\frac{\mathbf{b}_{1}}{\mathbf{b}}=\frac{\mathbf{c}_{1}}{\mathbf{c}}\]

Condition for Parallelism:

The line is parallel to the plane, if

\[\overline{\mathbf{b}}.\overline{\mathbf{n}}=0\]

aa₁ + bb₁ + cc₁ = 0