# 1.8 Intersection of two planes

We saw earlier that two arbitrary lines in ^{2} may intersect, be parallel, or coincide. In an analogous way, two arbitrary planes in ^{3} may intersect, be parallel, or coincide.

In general, if two distinct planes intersect, then the set of common points is a line that lies in both planes.

For example, the (*x*, *y*)-plane and the (*x*, *z*)-plane intersect in the *x*-axis, which lies in both planes.

Similarly, the planes *x* − *y* = 0 and *x* + *y* + *z* = 1 intersect in a line; the points (*x*, *y*, *z*) on this line all satisfy both of the equations

*x* − *y* = 0 and *x* + *y* + *z* = 1.

Two planes in ^{3} may be parallel, and so cannot intersect. For example, the plane with equation *z* = 0 is the (*x*, *y*)-plane, and the plane with equation *z* = 1 is a plane parallel to the (*x*, *y*)-plane, passing through the point (0, 0, 1) on the *z*-axis. These two planes do not intersect; every point in the plane *z* = 1 lies at distance 1 above the plane *z* = 0.

Finally, two planes may coincide. For example, the planes with equations

coincide, since the second equation is simply the first multiplied by the number .

In general, two planes are coincident if the equation of one can be rearranged to be a multiple of the equation of the other.

## Example 10

Determine whether the planes with equations *z* = 2 and *y* = −1 intersect, are parallel, or coincide. Illustrate your answer with a sketch.

Hint: You have already sketched these two planes in Exercise 8.

### Answer

From the sketches in Exercise 8, the two planes are clearly not parallel or coincident; hence they must intersect in a line that lies in both planes.