Two planes: line of intersection

Direction of the line

The line lies in both planes, so it is perpendicular to both normals n1 and n2. Its direction is n1 × n2.

If n1 × n2 = 0 the normals are parallel, so the planes are parallel (or the same plane) and there is no single line.

A point on the line

Choose a convenient value for one coordinate (often z = 0) and solve the two plane equations simultaneously for the other two. If that fails (the line is parallel to z = 0), try x = 0 or y = 0.

Then the line is r = (that point) + λ(n1 × n2).

Alternative: solve the two cartesian equations with z = λ as a parameter.

Checking your line

Substitute the general point of your line into both plane equations: each should hold for every λ. Also check (n1 × n2) · n1 = 0 and (n1 × n2) · n2 = 0.

The angle between two planes is not a listed F3 point; the F3 applications are the distance from a point to a plane, the line of intersection of two planes and the shortest distance between skew lines.

Worked example

Find a vector equation of the line of intersection of x + y + z = 6 and 2x - y + z = 3.

1. n1 × n2 = (1, 1, 1) × (2, -1, 1) = (1×1 - 1×(-1), 1×2 - 1×1, 1×(-1) - 1×2) = (2, 1, -3)
2. Set z = 0: x + y = 6 and 2x - y = 3, so x = 3, y = 3

Answer: r = (3i + 3j) + λ(2i + j - 3k)

Check yourself

1. Find a direction vector for the line of intersection of x + 2y - z = 4 and 2x - y + z = 1.
   • -i + 3j - 5k
   • i - 3j - 5k
   • 3i + j + 0k
   • 2i - 2j - k
   Show answer

   i - 3j - 5k. (1, 2, -1) × (2, -1, 1) = (2×1 - (-1)(-1), (-1)(2) - 1×1, 1×(-1) - 2×2) = (1, -3, -5).

2. For the same two planes, find the x-coordinate of the point on the line of intersection where z = 0. Give your answer as a fraction in its simplest form.
   Show answer

   6/5. x + 2y = 4 and 2x - y = 1. Doubling the second and adding: 5x = 6, so x = 6/5 (and y = 7/5).

3. The planes x - y + 2z = 3 and 2x + y - z = 4 meet in a line. Find the y-coordinate of the point on this line where x = 0.
   Show answer

   11. With x = 0: -y + 2z = 3 and y - z = 4. Adding gives z = 7, so y = 11. (The direction of the line is (1, -1, 2) × (2, 1, -1) = (-1, 5, 3).)

4. When do two planes fail to meet in a single line?
   • When their normals are perpendicular
   • When neither passes through the origin
   • When their normals are parallel
   • When their equations have different constants
   Show answer

   When their normals are parallel. Parallel normals mean parallel (or identical) planes, and then n1 × n2 = 0.

5. Where does the line of intersection of the planes z = 0 and x + y + z = 5 cross the x-axis? Give the x-coordinate.
   Show answer

   5. On the x-axis y = 0 and z = 0, so x = 5.

6. Why is n1 × n2 a direction for the line of intersection?
   • It is parallel to each normal
   • It passes through the origin
   • It is perpendicular to each normal, so it lies in each plane
   • It is the sum of the two normals
   Show answer

   It is perpendicular to each normal, so it lies in each plane. Any direction in a plane is perpendicular to that plane's normal. The vector product is perpendicular to both normals.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics, Further Mathematics and Pure Mathematics specification (issue 3), Unit FP3

Last updated 8 November 2026. Spotted a mistake? Tell us.