Lines and planes in three dimensions, and the two products that make the geometry tractable: the scalar product for angles and the vector product for perpendiculars.
Syllabus points covered
- Use the equation of a line in vector and Cartesian form
- Use the equation of a plane in vector, scalar product and Cartesian form
- Calculate the vector product and use it to find areas and normals
- Find the angle between two lines, a line and a plane, or two planes
- Find the distance from a point to a line or a plane, and between two skew lines
Common mistakes
- Confusing the scalar and vector products, or their geometric meanings.
- Concluding two lines intersect after checking only two of the three components.
- Using the angle between a line and the normal to a plane as the angle with the plane itself.
- Forgetting the modulus when finding the distance between two skew lines — the formula gives a signed value, and distance is never negative.
Worksheets
Use these as soon as you've learned the topic and need to practise it. Start with the worksheet, check yourself against the answers, and only then look at the step-by-step solutions.
- Practice worksheet
- Worksheet answers
- Step-by-step solutionsComing soon
Topical past papers
Move on to these once you've worked through the worksheets and are ready for harder, exam-style questions. Real past-paper questions on this topic, with the official mark scheme and full worked solutions.
- Topical questions
- Mark scheme
- Step-by-step worked solutionsComing soon
Looking for whole papers by session? All Further Mathematics past papers