## Three - dimensional Geometry

# Distance of a Point from a Plane

- The distance from the point P(x
_{1}, y_{1}, z_{1}) to the plane ax + by + cz + d = 0 is \frac{|ax_{1}+by_{1}+cz_{1}+d|}{\sqrt{a^{2}+b^{2}+c^{2}}} - The distance from the point D (0, 0, 0) to the plane ax + by + cz + d = 0 is \frac{|d|}{\sqrt{a^{2}+b^{2}+c^{2}}}
- The distance between the two parallel planes ax + by + cz + d
_{1}= 0 and ax + by + cz + d_{2}= 0 is \frac{|d_{1}-d_{2}|}{\sqrt{a^{2}+b^{2}+c^{2}}} - If (α, β, γ) is the foot of the perpendicular from (x
_{1}, y_{1}, z_{1}) to the plane ax + by + cz + d = 0, then \frac{\alpha-x_{1}}{a}=\frac{\beta-y_{1}}{b}=\frac{\gamma-z_{1}}{c}=\frac{-(ax_{1}+by_{1}+cz_{1}+d)}{a^{2}+b^{2}+c^{2}} - If (α, β, γ) is the image of the point (x
_{1}, y_{1}, z_{1}) with respect to the plane ax + by + cz + d = 0, then \frac{\alpha-x_{1}}{a}=\frac{\beta-y_{1}}{b}=\frac{\gamma-z_{1}}{c}=\frac{-2(ax_{1}+by_{1}+cz_{1}+d)}{a^{2}+b^{2}+c^{2}} - The ratio in which the plane ax + by + cz + d = 0 divides the line segment joining two points (x
_{1}, y_{1}, z_{1}) and (x_{2}, y_{2}, z_{2}) is = − π_{11}: π_{22}= − (ax_{1}+ by_{1}+ cz_{1}+ d) : (ax_{2}+ by_{2}+ cz_{2}+ d) - Two points A(x
_{1}, y_{1}, z_{1}) B(x_{2}, y_{2}, z_{2}) are on the same side of the plane ax + by + cz + d = 0 if \frac{ax_{1}+by_{1}+cz_{1}+d}{ax_{2}+by_{2}+cz_{2}+d} > 0 - Two points A(x
_{1}, y_{1}, z_{1}) B(x_{2}, y_{2}, z_{2}) are on the opposite side of the plane ax + by + cz + d = 0 if \frac{ax_{1}+by_{1}+cz_{1}+d}{ax_{2}+by_{2}+cz_{2}+d} < 0 - The angular bisectors 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 is \frac{a_{1}x+b_{1}y+c_{1}z+d_{1}}{\sqrt{a_1^2+b_1^2+c_1^2}}=\pm\frac{a_{2}x+b_{2}y+c_{2}z+d_{2}}{\sqrt{a_2^2+b_2^2+c_2^2}} - If d
_{1}d_{2}> 0, then bisector containing origin is positive sign bisector not containing origin is negative sign - If d
_{1}d_{2}> 0 and a_{1}a_{2}+ b_{1}b_{2}+ c_{1}c_{2}> 0

Acute angle bisector is negative sign

Obtuse angle bisector is positive sign - If d
_{1}d_{2}> 0 and a_{1}a_{2}+ b_{1}b_{2}+ c_{1}c_{2}< 0

Acute angle bisector is positive sign

Obtuse angle bisector is negative sign - The perpendicular distance from P(α, β, γ) to the line \frac{x-x_{1}}{l}=\frac{y-y_{1}}{m}=\frac{z-z_{1}}{n} is \frac{|(\alpha-x_{1},\beta-y_{1},\gamma-z_{1})\times(l, m, n)|}{\sqrt{l^{2}+m^{2}+n^{2}}}
- The image of (α, β, γ) with respect to \frac{x}{l}=\frac{y}{m}=\frac{z}{n} is (2pl − α, 2pm − β, 2pn − γ) where p = lα + mβ + nγ.
- The condition that the line \frac{x-x_{1}}{l}=\frac{y-y_{1}}{m}=\frac{z-z_{1}}{n} lies on the plane ax + by + cz + d = 0 is a
_{1}x +b_{1}y + c_{1}z + d = 0 and al + bm + cn = 0 - The length of projection of \vec{b} on a vector perpendicular to \vec{a} in the plane of \vec{a},\vec{b} is \frac{|\vec{a}\times\vec{b}|}{|\vec{a}|}
- The perpendicular distance from a point p to the line joining the points A, B is \frac{|\overrightarrow{AP}\times\overrightarrow{AB}|}{|\overrightarrow{AB}|}
- If \vec{a},\vec{b},\vec{c} are the position vectors of the points A, B, C respectively, then perpendicular distance from C to the line AB is \frac{|\overrightarrow{AC}\times\overrightarrow{AB}|}{|\overrightarrow{AB}|}=\frac{|\vec{b}\times\vec{c}+\vec{c}\times\vec{a}+\vec{a}\times\vec{b}|}{|\vec{b}-\vec{a}|}

### View the Topic in this video From 36:56 To 48:55

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1. The distance of a point whose position vector is \vec{a} from the plane \vec{r}\cdot\hat{n}=d is |d-\vec{a}\cdot\hat{n}|

2. The distance from a point (x_{1}, y_{1}, z_{1}) to the plane Ax + By + Cz + D = 0 is \begin{vmatrix}\frac{Ax_{1}+By_{1}+Cz_{1}+ D}{\sqrt{A^2+B^2+C^2}} \end{vmatrix}.