# Angle between two planes - Introduction

The angle between two planes is defined as the angle between their normals If θ is the angle between two planes, then so is 180 - θ. We shall take the acute angle as the angle between two planes

**Vector Form:** If r.n

_{1}=d

_{1}and r.n

_{2}=d

_{2}are the equation of two planes then angle between them is given by the equation

**If A**

Cartesian Form:

Cartesian Form:

_{1}x + B

_{1}y + C

_{1}z + D

_{1}=0 and A

_{2}x + B

_{2}y + C

_{2}z + D

_{2}=0 are the Cartesian equations of two planes and θ is the angle between them then

Condition for parallelism and perpendicularity

**1.**If the planes are parallel then

**2.**If the planes are perpendicular then A

_{1}A

_{2}+ B

_{1}B

_{2}+ C

_{1}C

_{2}=0

Coplanarity of Two Lines

**Vector Form:** If r = a_{1} + λb_{1} and r= a_{2} + μb_{2} are the equations of two lines then they are said to be coplanar if (a_{2}-a_{1}).(b_{1} x b_{2})=0**Cartesian Form:** If A(x_{1}, y_{1}, z_{1}) and B(x_{2}, y_{2}, z_{2}) are two points with the direction ratios of parallel vectors <a_{1}, b_{1}, c_{1}> and <a_{2}, b_{2}, c_{2}>, then the lines are said to be coplanar if

Distance of a point from a plane

**Vector Form:** If the equation of the plane is in the form r.N=d, where N is normal to the plane, then the perpendicular distance is

The length of perpendicular from origin O to the plane r.N=d is |d|/|N|**Cartesian Form:** If P(x

_{1}, y

_{1}, z

_{1}) be the given point with position vector a and Ax + By + Cz=D be the equation of the plane then the perpendicular distance from P to the plane is given by d=

Angle between a Line and a Plane

If r=a+λb be the equation of the line and r.n=d be the equation of the plane the angle between them is given by

**Example:** Find the distance of a point (2, 5, -3) from the plane 6x - 3y + 2z - 4=0

**Distance**

Solution:

Solution:

= 13/7

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