A-level Further Maths Year 13 · Topic 5
Vectors, matrices and proof
This topic brings together the most powerful tools in pure Further Maths.
We will find intersections, angles and distances between lines and planes, and use the vector product.
Then we find eigenvalues and eigenvectors, diagonalise matrices and extend proof by induction.
- 4 subtopics
- 53 questions
- Free: 5.1 notes and 10 questions
In this topic
- 5.1Lines and planes: intersections, angles and distancesFree below · 14 questions
- 5.2The vector productOpen in Full Marks · 13 questions
- 5.3Eigenvalues, eigenvectors and diagonalisationOpen in Full Marks · 13 questions
- 5.4Further proof by induction: matrices, divisibility and inequalitiesOpen in Full Marks · 13 questions
5.1 · Free notes
Lines and planes: intersections, angles and distances
Key points
- Intersection of two planes is a line: find its direction from the vector product of the normals, and a point by setting one coordinate (such as z = 0) and solving.
- Distance from a point to a plane n₁x + n₂y + n₃z = d: |n₁x₀ + n₂y₀ + n₃z₀ − d| / |n|.
- Distance from a point to a line: find the foot of the perpendicular (using the scalar product) or use |(p − a) × d|/|d|.
- Shortest distance between skew lines r = a + λd₁ and r = b + μd₂: |(b − a)·n| / |n| where n = d₁ × d₂.
- Angle between two planes = angle between normals; angle between a line and a plane = 90° − angle between the direction and the normal.
In more detail
- Worked example: distance from P(2, 1, −1) to the plane 2x − 2y + z = 4: |4 − 2 − 1 − 4|/3 = 3/3 = 1.
- Reflection in a plane: find the foot of the perpendicular from P to the plane; the image is the same distance beyond it.
- Exam tip: check whether two lines are parallel before using the skew-lines formula (if parallel, n = 0).
Practice
Lines and planes: intersections, angles and distances: 10 questions
Pick an answer to check it, or open the answer for the explanation.
1. How do you find the direction of the line where two planes meet?
- The vector product of their normals
- The sum of the normals
- The scalar product
- The difference of the d values
Show answer
A: The vector product of their normals. It is perpendicular to both normals.
2. What is the distance from (1, 2, 2) to the plane x + 2y + 2z = 0?
- 9
- 1
- 3
- 0
Show answer
C: 3. |1 + 4 + 4|/3 = 3.
3. What is the formula for the distance from (x₀, y₀, z₀) to n₁x + n₂y + n₃z = d?
- n₁x₀ + n₂y₀ + n₃z₀
- |n₁x₀ + n₂y₀ + n₃z₀ − d|/|n|
- |n|/d
- d/|n|
Show answer
B: |n₁x₀ + n₂y₀ + n₃z₀ − d|/|n|. Standard result.
4. For skew lines, n = d₁ × d₂ is used because …
- it is parallel to both
- it is the shortest line
- it is a unit vector always
- it is perpendicular to both lines
Show answer
D: it is perpendicular to both lines. The common perpendicular direction.
5. What is the shortest distance between skew lines?
- |(b − a)·n|/|n| with n = d₁ × d₂
- |b − a|
- |d₁ × d₂|
- 0
Show answer
A: |(b − a)·n|/|n| with n = d₁ × d₂. Project the joining vector onto the common perpendicular.
6. The angle between the normals of two planes is 70°. What is the acute angle between the planes?
- 20°
- 110°
- 70°
- 90°
Show answer
C: 70°. It is the angle between the normals (or its supplement).
7. What is the distance from the origin to 2x − y + 2z = 9?
- 9
- 3
- 1
- 4.5
Show answer
B: 3. 9/3.
8. How do you find a point on the line of intersection of two planes?
- Use the normals
- Take the origin
- Average the d values
- Set one coordinate (e.g. z = 0) and solve the two equations
Show answer
D: Set one coordinate (e.g. z = 0) and solve the two equations. Any convenient point.
9. Planes x + y + z = 1 and 2x + 2y + 2z = 5 are …
- parallel
- perpendicular
- identical
- intersecting in a line
Show answer
A: parallel. Normals are parallel, d values not in the same ratio.
10. What is |(3, 4, 0)|?
- 7
- 25
- 5
- 12
Show answer
C: 5. √(9 + 16).
Keep going in Full Marks
Full Marks has 14 questions on Lines and planes: intersections, angles and distances (4 more than here), set out in practice levels. You also get notes and 3D models for all 4 subtopics of Vectors, matrices and proof, plus games, timed exams and past papers.