# Plane

A plane is a surface such that if any two points on it are taken on it, the line segment joining them lies completely on the surface.

# Normal to plane

A line, perpendicular to a plane is called a normal to the plane. Clearly, every line lying in a plane is perpendicular to the normal to the plane.

**Equation of plane in different form**

- Plane passing through a given point and normal to a given vector:
**Vector form:**Equation of plane passing through the point and perpendicular to the vector is = 0 or =*d*; where*d*= (known as scalar product form of plane), where is position vector of any point on the plane.**Cartesian form:**Equation of plane passing thorough the point (*x*_{1},*y*_{1},*z*_{1}) and direction ratios of normal to the plane is*a*,*b*,*c*is*a*(*x*â€“*x*_{1}) +*b*(*y*â€“*y*_{1}) +*c*(*z*â€“*z*_{1}) = 0. - Plane normal to unit vector
*n***Ë†**and at a distance*d* - Plane passing through three given points:
**Vector from:**Vector form of equation of the plane passing through three points*A*,*B*,*C*having position vector respectively is .**Cartesian form:**Let plane is passing through points*A*(*x*_{1},*y*_{1},*z*_{1}),*B*(*x*_{2},*y*_{2},*z*_{2}), and*C*(*x*_{3},*y*_{3},*z*_{3}).*P*(*x*,*y*,*z*) be any point on the plane. - Plane that passes through a point
*A*with position vector and is parallel to the given vectors is = 0. - Equation of plane passing through the given point and line is = 0.
- Equation of plane passing through two parallel lines
**Intercept from of a plane:**Let the plane meets the coordinate axes at the points*A*(*a*, 0, 0),*B*(0,*b*, 0),*C*(0, 0,*c*). Then equation of plane is given by = 1.- Equation of a plane parallel to a given plane
*ax*+*by*+*cz*+*d*= 0 is*ax*+*by*+*cz*+*k*= 0; where*k*is any scalar. - Plane passing through the line of intersection of 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*a*_{1}*x*+*b*_{1}*y*+*c*_{1}*z*+*d*_{1}) +*Î»*(*a*_{2}*x*+*b*_{2}*y*+*c*_{2}*z*+*d*_{2}) = 0, where*Î»*âˆˆ*R.*

# Angle between the two planes

The angle between the two planes is defined as the angle between their normals.

Angle

*Î¸*between planes; or between the planes*a*_{1}*x*+*b*_{1}*y*+*c*_{1}*z**+**e*_{1}= 0 and*a*_{2}*x*+*b*_{2}*y*+*c*_{2}*z*+*e*_{2}= 0, where and is given by cos .Planes are perpendicular if = 0 and parallel if exists the scalar

*Î»*, such that .# Point of intersection of line and a plane

To find the point of intersection of the line

and the plane

*ax*+*by*+*cz*+*d*= 0.Consider any point on the line (
â‡’ It must satisfy the equation of plane
âˆ´
â‡’ (
â‡’

*x*=*rl*+*x*_{1},*y*=*mr*+*y*_{1},*z*=*nr*+*z*_{1}) which is on the plane.*a*(*x*_{1}+*lr*) +*b*(*y*_{1}+*mr*) +*c*(*z*_{1}+*nr*) +*d*= 0*ax*_{1}+*by*_{1}+*cz*_{1}+*d*) +*r*(*al*+*bm*+*cn*) = 0*r*=# Angle between a line and a plane

The angle between a line and a plane is the complement of the angle between the line and the normal to the plane. If

*Î±*,*Î²*,*Î³*be the direction ratios of the line and*ax*+*by*+*cz*+*d*= 0 be the equation of plane and*Î¸*be the angle between the line and the plane.â‡’ cos (90Â° â€“

*Î¸*) =or sin

*Î¸*=**Vector form**If

*Î¸*is the angle between the line; and plane

â‡’

If be parallel to plane

*ax*+*by*+*cz*+*d*= 0 iff;*Î¸*= 0 or*Ï€*or sin*Î¸*= 0 or*al*+*bm*+*cm*= 0.# Distance of a point from a plane

**Cartesian form**The length of perpendicular

*PM*from a point

*P*(

*x*

_{1},

*y*

_{1},

*z*

_{1}) to the plane

*ax*+

*by*+

*cz*+

*d*= 0 is given by

*PM*=

**Vector form**Distance of point from the plane is given by

*PM*= .

# Distance between the parallel planes

The distance between two parallel planes

*ax*+*by*+*cz*+*d*_{1}= 0 and*ax*+*by*+*cz*+*d*_{2}= 0 is given by**Equation of acute angle bisector of the planes**

Equation of acute angle bisector of 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 isif

*d*_{1},*d*_{2}> 0, and*a*_{1}*a*_{2}+*b*_{1}*b*_{2}+*c*_{1}*c*_{2}> 0Equation of obtuse angle bisector is

if

*d*_{1},*d*_{2}> 0, and*a*_{1}*a*_{2}+*b*_{1}*b*_{2}+*c*_{1}*c*_{2}< 0