Relative Position of Lines and Planes and Trace Lines
Two lines in space are parallel, coincident, intersecting or skew; a line and a plane are parallel, contained or intersecting; two planes are parallel, coincident or intersect in a line, and each case is decided by comparing direction and normal vectors with a triple product or a small linear system. In brief: for the lines with directions d1, d2 through A1, A2, compute d1 × d2. If it is zero, the lines are parallel or coincident. Otherwise, the lines intersect when (A2 − A1) · (d1 × d2) = 0 and are skew when it is not.
This guide gives a decision procedure for each pair of objects, with formulas for the distance between skew lines, the angle between a line and a plane, and the trace lines of a plane, with worked examples and a table.
How do two lines in space relate?
For L1 = A1 + t d1 and L2 = A2 + s d2, compute n = d1 × d2.
- If n = 0: parallel directions. The lines coincide if A2 lies on L1, otherwise they are parallel and distinct.
- If n ≠ 0 and (A2 − A1) · n = 0: the lines are coplanar and intersect in exactly one point.
- If n ≠ 0 and (A2 − A1) · n ≠ 0: the lines are skew, with no common point and no common plane. The distance between them is |(A2 − A1) · n|/|n|.
Example 1: intersecting lines
L1: (0, 0, 0) + t(1, 2, 3); L2: (1, 0, −1) + s(0, 1, 2). The directions are not parallel. Solve t = 1, 2t = s, 3t = −1 + 2s: t = 1, s = 2, and 3 = −1 + 4 ✓. They meet at (1, 2, 3).
Example 2: skew lines
L1: (1, 2, 3) + t(2, 1, −1); L2: (4, 1, 3) + s(1, 1, 1). n = (2, 1, −1) × (1, 1, 1) = (1 · 1 − (−1) · 1, (−1) · 1 − 2 · 1, 2 · 1 − 1 · 1) = (2, −3, 1). A2 − A1 = (3, −1, 0), and the dot product with n is 6 + 3 + 0 = 9 ≠ 0, so the lines are skew. The distance is 9/√14 ≈ 2.41.
Example 3: parallel lines
L1: t(1, 2, 3); L2: (1, 0, −1) + s(2, 4, 6). The directions are proportional. A2 = (1, 0, −1) is not on L1 (it would need t = 1 and y = 2). So the lines are parallel and distinct. The distance is |(A2 − A1) × d1|/|d1| = |(1, 0, −1) × (1, 2, 3)|/√14 = |(2, −4, 2)|/√14 = √24/√14 ≈ 1.31.
How do a line and a plane relate?
For a line A + t d and the plane n · X + D = 0, compute n · d.
- If n · d ≠ 0: the line pierces the plane at one point; substitute the parametric equations into the plane and solve for t.
- If n · d = 0 and A is in the plane: the line lies in the plane.
- If n · d = 0 and A is not in the plane: the line is parallel to the plane, at the distance |n · A + D|/|n|.
The angle α between a line and a plane satisfies sin α = |n · d|/(|n||d|), because it is the complement of the angle between d and the normal.
Example 4: a piercing point
The line (1, 2, 3) + t(1, −1, 2) and the plane x + y + z = 12. Then 6 + 2t = 12, t = 3, and the point is (4, −1, 9). The angle: n · d = 2, |n| = √3, |d| = √6, sin α = 2/√18 ≈ 0.4714, α ≈ 28.1°.
Example 5: parallel and contained
The direction d = (1, 1, 2) and the plane x + y − z = 5: n · d = 1 + 1 − 2 = 0. The line through (1, 2, 3) has 1 + 2 − 3 = 0 ≠ 5, so it is parallel, at the distance 5/√3 ≈ 2.89. For the plane x + y − z = 0, the same line lies in the plane.
How do two planes relate?
Compare the normals n1 and n2. If n1 × n2 = 0, the planes are parallel; they coincide if the whole equation is proportional (including D) and are distinct otherwise. If n1 × n2 ≠ 0, the planes meet in a line with the direction n1 × n2.
Example 6: intersecting planes
x + y + z = 3 and x − y = 0. The direction is (1, 1, 1) × (1, −1, 0) = (1, 1, −2). A common point: x = y and 2x + z = 3; choose x = 0, then z = 3, giving (0, 0, 3). The line: (t, t, 3 − 2t).
Example 7: parallel and coincident
x + y + z = 3 and x + y + z = 5 are parallel with the distance 2/√3 ≈ 1.15. The plane 2x + 2y + 2z = 6 coincides with the first.
Example 8: a system of three planes
Three planes meet in one point if the 3 × 3 determinant of their normals is nonzero. x + y + z = 6, x − y + z = 2, y + z = 5: the determinant of the normal matrix [[1, 1, 1], [1, −1, 1], [0, 1, 1]] is 1(−1 − 1) − 1(1 − 0) + 1(1 − 0) = −2 ≠ 0, so there is exactly one point: from the first two, y = 2 and x + z = 4; the third gives z = 3, x = 1, the point (1, 2, 3).
What are trace lines of a plane?
The trace lines (traces) of a plane are its intersections with the coordinate planes: set z = 0 for the trace on the xy-plane, y = 0 for the xz-plane and x = 0 for the yz-plane. The traces help to sketch the plane, since the portion of the plane in the first octant is a triangle bounded by them.
Example 9: traces of 6x + 3y + 2z = 6
- xy-plane (z = 0): 6x + 3y = 6, that is 2x + y = 2.
- xz-plane (y = 0): 6x + 2z = 6, that is 3x + z = 3.
- yz-plane (x = 0): 3y + 2z = 6.
The traces pass through the axis intercepts (1, 0, 0), (0, 2, 0) and (0, 0, 3), which are the vertices of the first-octant triangle.
Decision table
| Pair | Test | Outcome |
|---|---|---|
| two lines | d1 × d2 = 0 | parallel or coincident |
| two lines | (A2 − A1) · (d1 × d2) = 0 | intersecting |
| two lines | that product ≠ 0 | skew |
| line and plane | n · d ≠ 0 | one common point |
| line and plane | n · d = 0 | parallel or contained |
| two planes | n1 × n2 = 0 | parallel or coincident |
| two planes | n1 × n2 ≠ 0 | meet in a line |
Where does the classification come from?
The classification of the positions of lines and planes was part of Euclid’s solid geometry (Book XI of the Elements), where the notion of skew lines appears implicitly. The algebraic criteria with vectors and determinants were developed in the nineteenth century as part of linear algebra and analytic geometry.
Which mistakes are common?
Concluding that non-parallel lines intersect
In space, lines with different directions can be skew. Check the triple product.
Reusing a parameter
Use t for one line and s for the other. A common t forces the same moment, which is an extra restriction.
Missing the contained case
n · d = 0 does not mean the line is outside the plane. Test one point of the line.
Confusing the angle with its complement
The angle between a line and a plane uses sine of n · d; the cosine formula gives the angle with the normal instead.