## Three - dimensional Geometry

# Angle between Two Planes

- If 'θ' is an angle between the two planes \vec{r}\cdot \vec{n}_{1}=d_{1} and \vec{r}\cdot \vec{n}_{2}=d_{2}, then \cos \theta=\frac{\vec{n}_{1}\cdot\vec{n}_{2}}{|\vec{n}_{1}| \ |\vec{n}_{2}|}
- If θ is an angle between the two 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=\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}} - Plane || Plane ⇒ Normal || Normal
- Plane ⊥ Plane ⇒ Normal ⊥ Normal

### View the Topic in this video From 06:50 To 16:00

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1. In the vector form, if θ is the angle between the two planes, \vec{r}\cdot \vec{n}_{1}=d_{1} and \vec{r}\cdot \vec{n}_{2}=d_{2} then, \theta=\cos^{-1}\frac{|\vec{n}_{1} \cdot \vec{n}_{2}|}{|\vec{n}_{1}| \ |\vec{n}_{2}|}.

2. The angle θ 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 is given by \cos \theta=\begin{vmatrix}\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}} \end{vmatrix}