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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
  1. Plane passing through a given point and normal to a given vector:
     
    Vector form: Equation of plane passing through the point 87396.png and perpendicular to the vector 87310.png is 87319.png = 0 or 87325.png = d; where d = 87331.png (known as scalar product form of plane), where 87337.png is position vector of any point on the plane.
     
    Cartesian form: Equation of plane passing thorough the point (x1, y1, z1) and direction ratios of normal to the plane is a, b, c is a(xx1) + b(y y1) + c(zz1) = 0.
  2. Plane normal to unit vector nˆ and at a distance d from the origin: 87352.png.
  3. 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 87361.png respectively is 87367.png.
     
    Cartesian form: Let plane is passing through points A(x1, y1, z1), B(x2, y2, z2), and C(x3, y3, z3).
     
    Let P(x, y, z) be any point on the plane.
     
    Then vectors 87373.png are coplanar then 87383.png = 0.
     
    87389.png = 0
     
    which is the required equation of plane.
  4. Plane that passes through a point A with position vector 87417.png and is parallel to the given vectors 87410.pngis 87423.png = 0.
  5. Equation of plane passing through the given point 87431.png and line 87437.png is 87445.png = 0.
  6. Equation of plane passing through two parallel lines 87451.png and 87457.png is 87463.png = 0.
  7. 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 87504.png = 1.
  8. 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.
  9. Plane passing through the line of intersection of planes a1x + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 0 is (a1x + b1y + c1z + d1) + λ(a2x + b2y + c2z + d2) = 0, where λ R.

Angle between the two planes

The angle between the two planes is defined as the angle between their normals.
 
Angle θ between planes; 87513.png or between the planes a1x + b1y + c1z + e1 = 0 and a2x + b2y + c2z + e2 = 0, where 87519.png and 87525.png is given by cos 87537.png.
 
Planes are perpendicular if 87543.png = 0 and parallel if exists the scalar λ, such that 87549.png.

Point of intersection of line and a plane

To find the point of intersection of the line
 
87556.png
 
and the plane ax + by + cz + d = 0.
 
Consider any point on the line (x = rl + x1, y = mr + y1, z = nr + z1) which is on the plane.
 
It must satisfy the equation of plane
 
a(x1 + lr) + b(y1 + mr) + c(z1 + nr) + d = 0
 
(ax1 + by1 + cz1 + d) + r(al + bm + cn) = 0
 
r = 87574.png

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 α, β, γ be the direction ratios of the line and ax + by + cz + d = 0 be the equation of plane and θ be the angle between the line and the plane.
 
90826.png
 
cos (90° – θ) = 87580.png
or sin θ = 87586.png
 
Vector form If θ is the angle between the line; 87592.png and plane 87600.png
87606.png
 
If 87612.png be parallel to plane ax + by + cz + d = 0 iff; θ = 0 or π or sin θ = 0 or al + bm + cm = 0.

Distance of a point from a plane

Cartesian form The length of perpendicular PM from a point P(x1, y1, z1) to the plane ax + by + cz + d = 0 is given by
 
PM = 87619.png
 
Vector form Distance of point 87666.pngfrom the plane 87672.png is given by PM = 87649.png.

Distance between the parallel planes

The distance between two parallel planes ax + by + cz + d1 = 0 and ax + by + cz + d2 = 0 is given by
 
87658.png
 
Equation of acute angle bisector of the planes
 
Equation of acute angle bisector of the planes a1x + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 0 is
87678.png
 
if d1, d2 > 0, and a1a2 + b1b2 + c1c2 > 0
 
Equation of obtuse angle bisector is
 
87684.png
 
if d1, d2 > 0, and a1a2 + b1b2 + c1c2 < 0




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