The Angle between Planes

Consider the planes given by the equations

; ,

which have normal vectors and . Using inner product, we find the cosine of the angle:

.

The condition for two planes to be parallel coincides with the condition for the normal vectors and to be collinear:

.

The perpendicularity conditioin coincides with the perpendicularity condition for the vectors and :

( · )=0, i.e., A₁A₂+B₁B₂+C₁C₂= 0.

Example 1. Show that the following planes are parallel or perpendicular:

.

The planes are perpendicular, because

.

Example 2. Write an equation of the plane passing through the point М ₀(–1;2; 4) and parallel to the plane 6 x- 7 y+ 5 z+ 11 = 0.

The normal vector is normal to the required plane also. We have

A(x–x₀)+B(y–y₀)+C(z–z₀)= 0,

6 (x+ 1 )– 7 (y– 2 )+ 5 (z– 4 )= 0; 6 x– 7 y+ 5 z= 0.

The normal equation of a plane. Consider a plane. Let us draw the perpendicular ОР from the origin to this plane. Let a,b, and g be the angels between this perpendicular and the coordinate axes x,y, and z, and let . It is required to write an equation of this plane.

z

P

M(x,y,z)

0 y

x

 

 

Take an arbitrary point M(x; y; z) in the planeand consider the radius-vector . The unit vector on the perpendicular ОР has the coordinates

= { cos α; cos b; cos g}.

For any point М in the plane, the projection of the vector on the unit vector equals р:

.

Consider the scalar product

,

or, in coordinate form, .

Thus, we have obtained the normal equation of the plane:

 

. (25)

 

The normalizing factor. Consider the plane given by the general equation Ax+By+Cz+D= 0.

It is required to reduce this equation to the normal form (25).

Definition. Number m is called the normalizing factor if the equation multiplied by it is normal.

To find the normalizing factor, we multiply the general equation of the plane by a number m term by term:

mAx+mBy+mCz+mD= 0.

This equation is normal if the two normality conditions hold, i.e.,

1. (mA)²+(mB)²+(mC)²= 1,

2. mD< 0.

From the first condition, taking out m² and extracting the square root, we find the normalizing factor.

.

The sign opposite to that of the free term D must be taken.