Angle between two planes

Created: Friday, 16 September 2011 08:24 | Published: Friday, 16 September 2011 08:24 | Written by Super User | Print

Angle between two planes - Introduction    

ang_plane_1
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.n1=d1 and r.n2=d2 are the equation of two planes then angle between them is given by the equation
                       ang_plane_3

                             ang_plane_4

Cartesian Form:  If A1x + B1y + C1z + D1=0 and A2x + B2y + C2z + D2=0 are the Cartesian equations of two planes and θ is the angle between them then

                    ang_plane_5



Condition for parallelism and perpendicularity

1. If the planes are parallel then ang_plane_6
2. If the planes are perpendicular then A1A2 + B1B2 + C1C2=0



Coplanarity of Two Lines

Vector Form:  If r = a1 + λb1 and r= a2 + μb2 are the equations of two lines then they are said to be coplanar if (a2-a1).(b1 x b2)=0

Cartesian Form:  If A(x1, y1, z1) and B(x2, y2, z2) are two points with the direction ratios of parallel vectors <a1, b1, c1> and <a2, b2, c2>, then the lines are said to be coplanar if
         
               ang_plane_7



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 ang_plane_8

The length of perpendicular from origin O to the plane r.N=d is |d|/|N|

Cartesian Form:  If P(x1, y1, z1) 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= ang_plane_13


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
                              ang_plane_10

                               ang_plane_11


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

Solution: Distance ang_plane_12

                            ang_plane_2
                      
                            = 13/7



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