## Tuesday, May 20, 2008

### Revision Material 31. Three Dimensional Geometry - 3. Plane

Plane

A plane is a surface such that if any two points are taken on it, the line segment joining them lies completely on the surface. It means every point on the line segment joining any two points on the plane lies on the plane.

Every first degree equation in x,y, and z represents a plane.
ax + by + cz + d = 0 is the general equation of the plane.

We can also write it as

Ax + By + Cz +1 = 0; This is an equation in three unknowns.

Equation of plane passing through a given point

The general equation of a plane passing through a point (x1,y1, z1) is
a(x–x1)+b(y-y1)+c(z-z1) = 0, where a, b and c are constants.

Intercept form of plane

The equation of a plane intercepting lengths a, b an c with x-axis, y-axis and z-axis respectively is

x/a +y/b + z/c = 1

This plane intercepts lengths a, b, and c with x, y and z-axis respectively.

Given a plane equation, to find x-intercept, we put y =0 and z = 0. Similarly we do for other intercepts y and z.

Vector equation of a plane passing through a given point and normal to a given vector

The vector equation of a plane ‘Pi’ passing through a point having position vector a and normal to vector n is (r-a).n = 0 or r.n = a.n.

Where r is the position vector of an arbitrarily chosen point.

It can also be written as r.n = d
Where d = a.n

In Cartesian form

It is (x-a1)n1 + (y-a2)n2 +(z-a3n3) = 0

Where (a1,a2 and a3) are coordinates of the point a, and n1,n2 and n3 are the direction ratios of vector n normal to the plane.

Therefore, you can remember that coefficients of x,y and z in the Cartesian equation of a plane are the direction ratios of normal to the plane.

Equation of a plane in normal form

Vector form: the vector equation of a plane normal to unit vector n^ and at a distance d from the origin is r.n^ = d.

Cartesian form: Im l,m,n are direction cosines of the normal to a given plane which is at a distnance p from the origin, then the equation of the plane is:

lx + my +nz = p

The equation r.n^ = d denotes the distance of the plane from the origin.

Angle between two planes

The angle between two planes is defined as the angle between their normals.

Angle between planes in normal form:

Plane 1: r.n1 = d1.

Plane 2: r.n2 = d2.

cos θ = (n1.n2)/[ |n1| |n2|]

Condition of perpendicularity of planes

(n1.n2) = 0

Condition of parallelism of planes

n1 = λn2

Angle between two planes in Cartesian form

The angle θ between the planes a1x +b1y+c1z+d1 = 0, a2x +b2y+c2z+d2 = 0 is given by:

cos θ = [a1a2+b1b2+c1c2]/ [√[a1²+b1²+c1²] √[a2²+b2²+c2²]]

Condition of perpendicularity of planes
[a1a2+b1b2+c1c2] = 0

Condition of parallelism

a1/a2 =b1/b2 =c1/c2

Equation of a plane passing through a given point and parallel to two given vectors

Equation for the plane passing though a point having position vector a and parallel to b and c is r = a + λb + μc where λ and μ are scalars. This equation is called a parametric form of the plane as for different points on the plane the values of scalars λ and μ are different.

Nonparametric form: Equation for the plane passing though a point having position vector a and parallel to b and c is

(r-a).(b×c) = 0 or r.(b×c) = a.(b×c)

[rbc] = [abc]

Equation of a plane parallel to a given plane

Vector form: Parallel planes have the common normal. Hence equation of a plane parallel to the plane r .n = d1 is r.n = d2, where d2 is a constant determined by the given conditions.

Cartesian form: Let ax+by+cz+d = 0 be a plane. The direction ratios of its normal are a,b,c. A plane parallel to this plane will also have the same direction ratios. Hence the equation of the plane is ax+by+cz+k = 0 where k is an arbitrary constant and is etermined by the given conditions.

Equation of a plane passing through the intersection of two planes

Vector form: If the given two planes are r .n = d1 and r.n = d2, then the plane passing through the intersection of these two planes is:

(r.n1 – d1) + λ (r.n2 – d2) = 0

Or
r.(n1+ λn2) = d1+ λd2, where λ is an arbitrary constant.

Cartesian form:
Two planes give are:
a1x+b1y+c1z+d1 = 0
a2x+b2y+c2z+d2 = 0
The equation of the plan passing through the intersection of two planes is
(a1x+b1y+c1z+d1)+ λ (a2x+b2y+c2z+d2 = 0)