Plane, Equation of a Plane in the Intercept Form, Angle Between Two Planes, Distance of a Point from a Plane

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Equation of a Plane in the Intercept Form

Let be the lengths of the intercepts made by the plane on the and axes respectively.

It implies that the plane passes through the points and

Putting in (A),

We get the required equation of the plane as

Which on expanding gives or ……. (B)

Equation (B) is called the Intercept form of the equation of the plane.

Example:

Find the equation of the plane passing through the points and .

Solution:

Using (A), we can write the equation of the plane as

or

or

Angle between Two Planes:

Angle Between two planes

Angle between Two Planes

Angle Between two planes

Let the two planes and be given by

…… (1)

And …… (2)

Let the two planes intersect in the line and let and be normal to the two planes. Let be the angle between two planes.

The direction cosines of normals to the two planes are

And

is given by

Where the sign is so chosen that is positive

Corollary 1:

When the two planes are perpendicular to each other then i.e.,

The condition for two planes and to be perpendicular to each other is

Corollary 2:

If the two planes are parallel, then the normal to the two planes are also parallel

The condition of parallelism of two planes and

is

This implies that the equations of two parallel planes differ only by a constant. Therefore, any plane parallel to the plane is , where is a constant.Example:

Find the angle between the planes

…… (1)

And …… (2)

Solution:

Here

And

If is the angle between the planes (1) and (2), then

Thus the two planes given by (1) and (2) are perpendicular to each other.

Distance of a Point from a Plane

Let the equation of the plane in normal form be

where …… (1)

Case 1:

Let the point lie on the same side of the plane in which the origin lies.

Let us draw a plane through point parallel to plane (1).Its equation is

…… (2)

Where is the length of the perpendicular drawn from origin upon the plane given by (2). Hence the perpendicular distance of P from plane (1) is .

As the plane (2) passes through the point,

The distance of from the given plane is

Case 2:

If the point lies on the other side of the plane in which the origin lies, then the distance of from the plane (1) is,

Example:

Find the distance of the point from the plane

Solution:

Required distance

units.

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