# 1.7 Planes in three-dimensional Euclidean space

We now look at the general form of the equation of a plane in ^{3}.

Three planes whose equations are easy to find are those that contain a pair of axes. For example, the (*x*, *y*)-plane is the plane that contains the *x*-axis and the *y*-axis. Points which lie in this plane are precisely those points (*x*, *y*, *z*) in ^{3} for which *z* = 0, so the equation of the (*x*, *y*)-plane is *z* = 0.

## Example 7

Find the equations of the (*y*, *z*)-plane and the (*x*, *z*)-plane.

### Answer

Points (*x*, *y*, *z*) that lie in the (*y*, *z*)-plane all have *x* = 0; so *x* = 0 is the equation of this plane.

Similarly, points (*x*, *y*, *z*) that lie in the (*x*, *z*)-plane all have *y* = 0; so *y* = 0 is the equation of this plane.

## Example 8

Sketch the planes whose equations are as follows.

**(a)***z*= 2**(b)***y*= −1

### Answer

**(a)****(b)**

You saw in Section 1.2 that the general form of the equation of a line in ^{2} is *ax* + *by* = *c*, where *a*, *b* and *c* are real, and *a* and *b* are not both zero. In ^{3} the analogue of this equation is the equation of a plane.

We have already met the planes with equations *x* = 0, *y* = 0 and *z* = 0. Each of these equations is a special case of a more general equation
*ax* + *by* + *cz* = *d *,
where *a*, *b*, *c* and *d* are real, and *a*, *b* and *c* are not all zero. In fact, the equation of any plane in ^{3} is of this form.

## Equation of a plane

The general equation of a plane in ^{3} is *ax* + *by* + *cz* = *d*, where *a*, *b*, *c* and *d* are real, and *a*, *b* and *c* are not all zero.

(We shall prove this in Section 3.3).

## Example 3

Determine the general form of the equation of a plane that passes through the origin.

### Answer

Let the plane have equation

*ax* + *by* + *cz* = *d*

Since (0, 0, 0) lies in the plane, its coordinates must satisfy the equation of the plane; thus

*a* × 0 + *b* × 0 + *c* × 0 = *d*,

so *d* = 0. Also, the plane whose equation is

*ax* + *by* + *cz* = 0

clearly passes through the origin (0, 0, 0).

Hence the general form of the equation of a plane that passes through the origin is

*ax* + *by* + *cz* = 0

where *a*, *b* and *c* are real and not all zero.

The (*x*, *y*)-, (*y*, *z*)- and (*x*, *z*)-planes all pass through the origin, and their equations are all of the form shown above.

## Example 9

For each of the following points, determine the general form of the equation of a plane that passes through the point.

**(a)**(1, 2, 3)**(b)**(−1, −4, 2)

### Answer

**(a)**Let the equation of the plane be of the form*ax*+*by*+*cz*=*d*Since (1, 2, 3) lies in the plane, its coordinates must satisfy the equation of the plane; thus

*a*× 1 +*b*× 2 +*c*× 3 =*d*,so

*d*=*a*+ 2*b*+ 3*c*.Thus the general form of the equation of a plane through (1, 2, 3) is

*ax*+*by*+*cz*=*a*+ 2*b*+3*c***(b)**Let the equation of the plane be of the form*ax*+*by*+*cz*=*d*Since (−1, −4, 2) lies in the plane, its coordinates must satisfy the equation of the plane; thus

*a*× (−1) +*b*× (−4) +*c*× 2 =*d*,so

*d*= −*a*- 4*b*+ 4*c*.Thus the general form of the equation of a plane through (−1, −4, 2) is

*ax*+*by*+*cz*= −*a*− 4*b*+ 2*c*.