A line and a plane: every case

Choose your method, connect the parts and explain your conclusions. These original questions include written reasoning and sketches as well as exact answer checks.

Suggested marks guide how much working to show. The site does not automatically award examination marks for proofs, diagrams or methods. Use scaffolded fluency practice when you need a hint first.

A-level core. Course filters use selected objective mappings; none of these tasks establishes complete paper coverage.

Before you start

Scalar products and angles. Read the relevant method, then decide which parts you can solve without hints.

Check a misconception first · Try a connected problem

Read an original example question

The plane is x + y = 4. A line has vector equation r = (0, 4, 2) + t(1, −1, 3).

Part a

Find the scalar product of a plane normal and the line direction.

Part b

State and justify the relationship between the line and plane.

Part c

Explain how the conclusion changes if the line direction becomes (1, 0, 3).

Model solution and review criteria

Part a

A normal is (1,1,0), whose scalar product with (1,−1,3) is zero.

Select a plane normal and use all components.

Part b

Every point has x+y = t + 4−t = 4. The whole line lies in the plane.

Use the constant x+y along the line.
Distinguish contained from parallel-disjoint.

Part c

Now x+y = t + 4. There is exactly one intersection at t = 0. The normal-direction product is 1, so division is valid.

Recompute the normal-direction product.
Identify a unique parameter and intersection.

Compare your own reasoning. No automatic examination marks are awarded.