Picture a pencil pushed through a sheet of paper: A line is a moving point: r = a + λd, so its coordinates are x = a₁ + λd₁, y = a₂ + λd₂, z = a₃ + λd₃ — all written in terms of one parameter λ.
A plane is all points (x, y, z) that satisfy one equation like n₁x + n₂y + n₃z = k (its Cartesian form, where (n₁, n₂, n₃) is the normal).
The meeting point is on BOTH. So put the line's x, y, z into the plane's equation — you get one equation in just λ. Solve it, then feed λ back into the line.
IB-style question — find the intersection point
A line has equation r = (1, 0, 2) + λ(2, 1, −1). A plane has equation x + 2y + z = 9.
Find the coordinates of the point where the line meets the plane.
Step by step
- Write the line's coordinates in terms of λ — these are the (x, y, z) of any point on the line.
- The meeting point also obeys the plane's equation, so substitute these in.
- Expand and collect the λ-terms and the constants.
- Solve the single equation for λ.
- Put λ = 2 back into the line to get the actual point.
Final answer
The line meets the plane at (5, 2, 0).
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When the parameter vanishes, something special is happening: After substituting, you normally get (number)·λ = number and solve for one λ. But sometimes the λ-terms cancel and you're left with just numbers:
• (false statement, e.g. 0 = 7): the line never satisfies the plane → it is parallel to the plane and misses it (no intersection).
• (true statement, e.g. 0 = 0): EVERY λ works → the line lies entirely in the plane (infinitely many common points).
Why the λ-terms cancel: the line's direction d is perpendicular to the plane's normal n (so d·n = 0). The line runs along the plane's surface direction; whether it sits in the plane or floats above it is decided by the constants.
IB-style question — show the line is parallel and misses
Show that the line r = (0, 1, 0) + λ(1, 1, 1) does not meet the plane x − 2y + z = 4.
Step by step
- The line's coordinates in terms of λ.
- Substitute into the plane's equation.
- Expand — watch the λ-terms.
- The λ-terms cancel (λ − 2λ + λ = 0), leaving an impossible statement.
- No λ can make this true, so there is no point on both.
Final answer
The statement −2 = 4 is impossible, so the line is parallel to the plane and never meets it.
IB-style question — show the line lies in the plane
Show that the line r = (2, 0, 1) + λ(1, 1, 1) lies in the plane x − 2y + z = 3.
Step by step
- The line's coordinates in terms of λ.
- Substitute into the plane's equation.
- Expand and collect.
- True for EVERY value of λ, so every point of the line is on the plane.
Final answer
The equation reduces to 3 = 3, true for all λ, so the whole line lies in the plane.