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Math101learn.math101.caIntersections of Lines and Planes
Line-plane and plane-plane intersections are found by combining parametric and scalar equations and interpreting the resulting system.
Algebraic systems reveal whether spatial objects meet once, never meet, or share infinitely many points.
Line-plane possibilities
A line and a plane can:
- intersect at one point;
- be parallel and disjoint;
- lie entirely in the plane.
The line direction $\vec d$ and plane normal $\vec n$ predict the structure. If $\vec d\cdot\vec n\ne0$, the line crosses the plane once. If the dot product is zero, the line is parallel to the plane and may either be disjoint or contained.
Substitution method
Write the line parametrically:
Substitute these expressions into the scalar plane equation. Solve for $t$, then return to the line equations to find the intersection point.
Worked example: one intersection
Parallel or contained line
If substitution simplifies to a contradiction such as $3=6$, no parameter works and the line is parallel but disjoint.
If it simplifies to an identity such as $0=0$, every parameter works and the entire line lies in the plane. Test one line point to distinguish these cases after confirming parallel direction.
Two planes
Two nonparallel planes intersect in a line. A direction vector for that line is
because the intersection direction lies in both planes and is perpendicular to both normals.
Find one point satisfying both plane equations, then combine it with $\vec d$ to write the intersection line.
Parallel and coincident planes
If normals are scalar multiples, the planes are parallel or coincident. Compare the full scalar equations:
- proportional coefficients and constants: same plane;
- proportional normal coefficients but inconsistent constants: distinct parallel planes.
There is no single intersection line in either case.
Three-plane systems
Three plane equations form a linear system in $x,y,z$. Possible solution sets include:
- one point;
- no common point;
- a common line;
- the same plane or another infinite family.
Use elimination or matrices, then interpret a unique solution, contradiction, or free variable geometrically.
Line-line intersections in space
Give the two lines different parameters and equate $x$, $y$, and $z$ components. A consistent pair of parameter values produces an intersection.
Nonparallel lines with an inconsistent system are skew, not parallel. Checking only a two-dimensional projection can falsely suggest an intersection.
Angles at intersections
Use direction vectors for angles between lines and normal vectors for angles between planes. For a line and plane, the acute angle $\phi$ satisfies
This angle is complementary to the angle between the line direction and plane normal.
Verification
An intersection point must satisfy every original equation. Substitute it into the line using its parameter and into each plane's scalar equation.
For an intersection line, verify its base point lies on both planes and its direction dots to zero with both normals.
Common mistakes
Substituting a direction vector as though it were a point. Use full parametric coordinates.
Finding $t$ but not the point. Return to all three line equations.
Calling an identity “one solution.” It means every line point lies in the plane.
Assuming nonintersecting spatial lines are parallel. They may be skew.
Using $\vec n_1\times\vec n_2$ when normals are parallel. The zero cross product cannot provide an intersection direction.
Quick self-check
- What geometric cases are possible before solving?
- Does $\vec d\cdot\vec n$ predict crossing or parallelism?
- Was one shared parameter substituted into the plane?
- Does the solved point satisfy every original equation?
- For two planes, is the cross-product direction nonzero and perpendicular to both normals?
- Has a unique, inconsistent, or infinite system been interpreted geometrically?
Related topics
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Where does r = ⟨1, 0, 2⟩ + t⟨2, 1, −1⟩ meet x + y + z = 6?
- 3 + 2t = 6, so t = 3/2.
- Substitute into the line: x = 4, y = 3/2, z = 1/2.
- The coordinates sum to 6.
End of lesson
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