Plane, Objectives, Vector Equation of a Plane, Equation of Plane in Normal from, Conversion of Vector Form into Cartesian Form

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If we consider any two points in a plane, the line joining these points will lie entirely in the same plane. This is the characteristic of a plane.

A representation of a rectangular box

A Representation of a Rectangular Box

A representation of a rectangular box

Look at Fig. You know that it is a representation of a rectangular box. This has six faces, eight vertices and twelve edges.

The pairs of opposite and parallel faces are

  1. and

  2. and

  3. and

And the sets of parallel edges are given below:

  1. and

  2. and

  3. and

Objectives:

After studying this lesson, you will be able to:

After studying this lesson

After Studying this Lesson

After studying this lesson

Vector Equation of a Plane:

A plane is uniquely determined if any one of the following is known:

  1. Normal to the plane and its distance from the origin is given.

  2. One point on the plane is given and normal to the plane is also given.

  3. It passes through three given non collinear points.

Equation of Plane in Normal From:

Let the distance of the plane from origin be and let be a unit vector normal to the plane. Consider as position vector of an arbitrary point on the plane.

Equation of Plane in Normal From

Equation of Plane in Normal From

Equation of Plane in Normal From

Since is the perpendicular distance of the plane from the origin and is a unit vector perpendicular to the plane.

Now

is perpendicular to the plane and lies in the plane, therefore

i.e.

i.e.

i.e. …… (1)

Which is the equation of plane in vector from.

Conversion of Vector Form into Cartesian Form:

Let be the co-ordinates of the point and be the direction cosines of .

Then

Substituting these value in equation (1) we get

This is the corresponding Cartesian form of equation of plane in normal form.

Example:

Find the distance of the plane from the origin. Also find the direction cosines of the unit vector perpendicular to the plane.

Solution:

The given equation can be written as

Dividing both sides of given equation by we get

i.e.

D.c.s of unit vector normal to the plane are and distance of plane from origin

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