Vectors and conics

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

xy = 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.

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.

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