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Equation of a Plane in Different Forms

Definition:   Consider the locus of a point P(x, y, z). If x, y, z are allowed to vary without any restriction for their different combinations, we have a set of points like P. The surface on which these points lie, is called the locus of P. It may be a plane or any curved surface. If Q be any other point on it’s locus and all points of the straight line PQ lie on it, it is a plane. In other words if the straight line PQ, however small and in whatever direction it may be, lies completely on the locus, it is a plane, otherwise any curved surface.

Equation of Plane in Different Forms:

  • General equation of a plane is ax + by + cz + d = 0

  • Equation of the plane in Normal form is lx + my + nz = p where p is the length of the normal from the origin to the plane and (l, m, n) be the direction cosines of the normal.

  • The equation to the plane passing through P(x1, y1, z1) and having direction ratios

    (a, b, c) for its normal is a(x – x1) + b(y – y1) + c (z – z1) = 0

  • The equation of the plane passing through three non-collinear points (x1, y1, z1),

    (x2, y2, z2) and (x3, y3 , z3) is 872_non-collinear points.JPG = 0

  • The equation of the plane whose intercepts are a, b, c on the x, y, z axes respectively is x/a + y/b + z/c = 1 (a b c ≠ 0)

  • Equation of YZ plane is x = 0,           equation of plane parallel to YZ plane is x = d.

           Equation of ZX plane is y = 0,           equation of plane parallel to ZX plane is y = d.

           Equation of XY plane is z = 0,           equation of plane parallel to XY plane is z = d.

  • Four points namely A (x1, y1, z1), B (x2, y2, z2), C (x3, y3, z3) and D (x4, y4, z4) will be coplanar if one point lies on the plane passing through other three points.

CXmhaU -8: Find the equation to the plane passing through the point (2, -1, 3) which is the foot of the perpendicular drawn from the origin to the plane.        

hb :     The direction ratios of the normal to the plane are 2, -1, 3.

                        The equation of required plane is 2(x –2) –1 (y + 1) + 3 (z –3) = 0

                        => 2x – y + 3z –14 = 0

Angle between the Planes

Angle between the planes is defined as angle between normals of the planes drawn from any point to the planes.

Angle between the planes a1x + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 0 is 268_Angle between planes.JPG

 

Note:

•         If a1a2 +b1b2 +c1c2 = 0, then the planes are perpendicular to each other.

•         If a1/a2 = b1/b2 = c1/c2 then the planes are parallel to each other.

CXmhaU -9: Find angle between the planes 2x – y + z = 11 and x + y + 2z = 3.

hb :    1987_Angle between planes.JPG

1254_Angle between planes.JPG

CXmhaU -10:           Find the equation of the plane passing through (2, 3, –4), (1, –1, 3) and parallel to x-axis.

hb :     The equation of the plane passing through (2, 3, –4) is

                        a(x – 2) + b(y – 3) + c(z + 4) = 0                              ……(1)

                        since (1, –1, 3) lie on it, we have

                        a + 4b – 7c = 0                                                          ……(2)

                        since required plane is parallel to x-axis i.e. perpendicular to YZ plane i.e.

                        1.a + 0.b + 0.c = 0 Þ a = 0 Þ 4b – 7c = 0 => b/7 = c/4

                         Equation of required plane is 7y + 4z = 5.

Perpendicular Distance:

The length of the perpendicular from the point P(x1, y1, z1) to the plane ax + by + cz + d = 0 is|ax1 + by1 + cz1 + d / √a2 + b2 + c2|.

Family of Planes:

Equation of plane passing through the line of intersection of two planes u = 0 and v = 0 is 
u + λv = 0.

Intersection of a Line and Plane:

If equation of a plane is ax + by + cz + d = 0, then direction cosines of normal to this plane are a, b, c. So angle between normal to the plane and a straight line having direction cosines l, m ,n is given by cos θ = al + bm + cn / √a2 + b2 + c2.

Then angle between the plane and the straight line is π/2 – θ.

  • Plane and straight line will be parallel if al + bm + cn = 0

  • Plane and straight line will be perpendicular if a/l = b/m = c/n.

CXmhaU -11:           Find the equation of plane passing through the intersection of planes 2x – 4y + 3z + 5= 0, x + y + z = 6 and parallel to straight line having direction cosines (1, –1, –1).

hb :     Equation of required plane be

                        (2x – 4y + 3z + 5) + λ(x + y – z – 6) = 0

                        i.e. (2 + λ)x + (–4 + λ)y + z(3 – λ) + (5 – 6λ)     = 0

                        This plane is parallel to a straight line. So, al + bm + cn = 0

                        1(2 + λ) + (–1)(–4 + λ) + (–1)(3 – λ) = 0 i.e. λ = –3

                         Equation of required plane is –x – 7y + 6z + 23 = 0

                        i.e. x + 7y – 6z – 23 = 0.

Bisector Planes of Angle between two Planes: 

The equation of the planes bisecting the angles between two given planes a1x +b1y +c1z +d1 = 0 and a2x + b2y + c2z +d2 = 0 is

1249_bisecting angles.JPG

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