A-level Further Maths Year 12 · Topic 5
Vectors and volumes
Vectors describe lines and planes in three dimensions.
We will write vector equations of lines, use the scalar product to find angles, and meet the equation of a plane.
Then we rotate curves to find volumes of revolution.
- 4 subtopics
- 86 questions
- Free: 5.1 notes and 10 questions
In this topic
- 5.1Vector equations of lines in three dimensionsFree below · 10 questions
- 5.2The scalar product: angles between vectors and lines, and perpendicularityOpen in Full Marks · 12 questions
- 5.3Equations of planes and the intersection of a line and a planeOpen in Full Marks · 13 questions
- 5.4Volumes of revolution about the x- and y-axesOpen in Full Marks · 51 questions
5.1 · Free notes
Vector equations of lines in three dimensions
Key points
- The vector equation of a line through point A (position vector a) with direction d is r = a + λd, where λ is a scalar parameter.
- Through two points A and B: r = a + λ(b − a).
- In Cartesian form: (x − a₁)/d₁ = (y − a₂)/d₂ = (z − a₃)/d₃ = λ.
- Two lines in 3D can be parallel (directions are multiples), intersecting (a common point satisfies both equations), or skew (neither parallel nor meeting).
- To test for intersection, set the equations equal, solve two of the component equations for λ and μ, and check the third.
In more detail
- Worked example: l₁: r = (1, 2, 0) + λ(1, 0, 1), l₂: r = (2, 1, 3) + μ(0, 1, 1). x: 1 + λ = 2 → λ = 1; y: 2 = 1 + μ → μ = 1; z: λ = 3 + μ? 1 ≠ 4, so the lines are skew.
- Shortest distance link: at A level you find the shortest distance between skew lines using the vector product.
- Exam tip: use different parameters (λ and μ) for different lines.
Practice
Vector equations of lines in three dimensions: 10 questions
Pick an answer to check it, or open the answer for the explanation.
1. What is the vector equation of the line through (1, 2, 3) with direction (2, 0, −1)?
- r = (1, 2, 3) + λ(2, 0, −1)
- r = (2, 0, −1) + λ(1, 2, 3)
- r = λ(1, 2, 3)
- r = (1, 2, 3) × (2, 0, −1)
Show answer
A: r = (1, 2, 3) + λ(2, 0, −1). r = a + λd.
2. Which point lies on r = (1, 0, 2) + λ(1, 1, 1)?
- (2, 2, 2)
- (0, 1, 2)
- (3, 2, 4)
- (1, 1, 1)
Show answer
C: (3, 2, 4). λ = 2.
3. What is the direction of the line through A(1, 1, 0) and B(4, 3, 2)?
- (5, 4, 2)
- (3, 2, 2)
- (1, 1, 0)
- (4, 3, 2)
Show answer
B: (3, 2, 2). b − a.
4. What are skew lines?
- Parallel lines
- Perpendicular lines that meet
- Lines through the origin
- Lines that are not parallel and do not meet
Show answer
D: Lines that are not parallel and do not meet. Only possible in 3D.
5. Lines with directions (2, −4, 6) and (−1, 2, −3) are …
- parallel
- perpendicular
- skew
- intersecting at O
Show answer
A: parallel. One is −2 times the other.
6. Write r = (2, −1, 3) + λ(4, 5, 1) in Cartesian form.
- (x − 4)/2 = (y − 5)/(−1) = (z − 1)/3
- (x + 2)/4 = (y − 1)/5 = (z + 3)/1
- (x − 2)/4 = (y + 1)/5 = (z − 3)/1
- x/4 = y/5 = z
Show answer
C: (x − 2)/4 = (y + 1)/5 = (z − 3)/1. (x − a₁)/d₁ form.
7. How many component equations do you check when testing for intersection?
- Only one
- Solve with two, check the third
- All four
- None
Show answer
B: Solve with two, check the third. Three equations, two unknowns.
8. What does it mean if the third equation fails?
- The lines intersect
- The lines are parallel
- They are the same line
- The lines are skew (if not parallel)
Show answer
D: The lines are skew (if not parallel). No common point.
9. What is the parameter λ for the point (5, 4, 6) on r = (1, 0, 2) + λ(2, 2, 2)?
- 2
- 4
- 1
- 3
Show answer
A: 2. 1 + 2λ = 5.
10. Can two lines in 2D be skew?
- Yes
- Only if parallel
- No
- Only if perpendicular
Show answer
C: No. In a plane, non-parallel lines always meet.
Keep going in Full Marks
Full Marks has 10 questions on Vector equations of lines in three dimensions, set out in practice levels. You also get notes and 3D models for all 4 subtopics of Vectors and volumes, plus games, timed exams and past papers.