Distance of a Point from a Plane

• The distance from the point P(x1, y1, z1) 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 + d1 = 0 and ax + by + cz + d2 = 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 (x1, y1, z1) 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 (x1, y1, z1) 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 (x1, y1, z1) and (x2, y2, z2) is = − π11 : π22 = − (ax1 + by1 + cz1 + d) : (ax2 + by2 + cz2 + d)
• Two points A(x1, y1, z1) B(x2, y2, z2) 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(x1, y1, z1) B(x2, y2, z2) 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 a1x +b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 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 d1d2 > 0, then bisector containing origin is positive sign bisector not containing origin is negative sign
• If d1d2 > 0 and a1a2 + b1b2 + c1c2 > 0
Acute angle bisector is negative sign
Obtuse angle bisector is positive sign
• If d1d2 > 0 and a1a2 + b1b2 + c1c2 < 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 a1x +b1y + c1z + 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

Disclaimer: Compete.etutor.co may from time to time provide links to third party Internet sites under their respective fair use policy and it may from time to time provide materials from such third parties on this website. These third party sites and any third party materials are provided for viewers convenience and for non-commercial educational purpose only. Compete does not operate or control in any respect any information, products or services available on these third party sites. Compete.etutor.co makes no representations whatsoever concerning the content of these sites and the fact that compete.etutor.co has provided a link to such sites is NOT an endorsement, authorization, sponsorship, or affiliation by compete.etutor.co with respect to such sites, its services, the products displayed, its owners, or its providers.

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 (x1, y1, z1) 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}.