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.n1=d1 and r.n2=d2 are the equation of two planes then angle between them is given by the equation
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
Condition for parallelism and perpendicularity
1. If the planes are parallel then 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
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(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=
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
Solution: Distance
= 13/7
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