About this chapter
Vectors closes Pure 4 with lines in three dimensions. Most questions follow a pattern: write the equation of a line, decide whether two lines meet, find the angle between them, then locate a special point such as the foot of a perpendicular or a reflection. The January 2024 examiners saw improvement on this topic, and also the same recurring errors with direction vectors.
This chapter covers vectors and distances, vector equations of lines, points on a line, intersecting, parallel and skew lines, the scalar product and angles, perpendiculars, feet and reflections, areas of triangles and parallelograms, and distances along a line.
The ten questions
- Q1 (6 marks): two vectors from a triangle's vertices, the angle between them, and which angle is largest.
- Q2 (5 marks): the line through two points, a point on it with unknown p and q, and a unit vector.
- Q3 (6 marks): two lines that intersect: find the constant q, then the point where they meet.
- Q4 (5 marks): prove two lines are skew.
- Q5 (6 marks): OP² as a quadratic in λ, then the closest point to O by completing the square.
- Q6 (10 marks): the fourth vertex of a parallelogram, an exact cosine, its area, and a shortest distance.
- Q7 (10 marks): the foot of the perpendicular from a point to a line, its reflection, and a triangle's area.
- Q8 (11 marks): where two lines meet, show cos θ = 4/9, then equal distances along each line.
- Q9 (12 marks): diagonals of a cuboid: the angle between them, where they meet, and a triangle's area.
- Q10 (11 marks): points C on a line with angle ACB = 90°, found from a quadratic and explained with a sphere.
Key skills tested
Vectors and distance. The vector from A to B is b − a, the end minus the start. The length of xi + yj + zk is √(x² + y² + z²), and a unit vector is a vector divided by its length.
Equation of a line. r = a + λd: a point on the line plus a multiple of the direction. Always write "r =".
Points on a line. Set a + λd equal to the point. One component gives λ, and the others must agree.
Intersecting lines. Equate the two lines, solve two components for λ and μ, then use the third to check, or to find a constant.
Parallel or skew. Parallel lines have directions that are multiples. Skew lines are not parallel and do not meet. State both, with reasons.
Scalar product and angles. cos θ = a·b ÷ (|a||b|), with both vectors pointing out of the angle. A negative value means an obtuse angle.
Perpendiculars. Perpendicular vectors have a·b = 0. For the foot X from P, use the vector from P to X dotted with the direction. The reflection of P is at 2x − p.
Areas. A triangle is ½|a||b| sin θ and a parallelogram |a||b| sin θ, with the exact sin θ = √(1 − cos²θ).
Distances along a line. |λd| = |λ||d|, so a given distance along a line gives two points, one each side. Draw a diagram.
Worked example
Question 2 from this chapter. A is (2, −1, 3) and B is (4, 3, −1). (a) Find a vector equation of the line l through A and B. (b) The point C(p, 9, q) lies on l: find p and q. (c) Find a unit vector parallel to l.
(a) The direction AB is 2i + 4j − 4k, or the simpler i + 2j − 2k. So l is r = (2i − j + 3k) + λ(i + 2j − 2k).
(b) The j components give −1 + 2λ = 9, so λ = 5. Then p = 2 + 5 = 7 and q = 3 − 10 = −7.
(c) The direction has length √(1 + 4 + 4) = 3, so a unit vector is (1/3)i + (2/3)j − (2/3)k, or its negative.
The "r =" at the start of the line's equation is needed for the accuracy mark.
Seen on real papers
Every question in our booklets is original. These are the recent papers where each question type has appeared.
- January 2024, Q6: two intersecting lines with an unknown constant, their meeting point, the angle between them, and a perpendicular condition, as in Q3, Q7 and Q8.
Where marks are lost
- Position vectors instead of directions. Using position vectors, often of the meeting point, to find the angle between two lines was fairly common in January 2024.
- Long routes to λ and μ. Substitution methods full of fractions led to slips, when solving two components directly was quicker.
- The acute angle to the wrong accuracy. Some rounded the angle to 2 significant figures or gave the obtuse angle instead.
- The wrong vector for a perpendicular. Many used a position vector instead of the direction from the fixed point to the line when finding the foot.
Common questions
How do I prove two lines are skew?
Show the directions are not multiples, so the lines are not parallel, then show the equations for λ and μ are inconsistent, so they do not meet. State both conclusions.
Which vectors go into the angle formula?
The direction vectors of the lines, or vectors pointing out of the angle, never position vectors.
How close does a line come to a point?
Find the foot of the perpendicular from the point, then measure from the point to that foot, or complete the square as in Q5.
Where this chapter leads
- Mechanics 1 Chapter 3: Vectors in Mechanics: vectors for velocity, position and forces.
- Pure 4 Chapter 1: Proof: proving results about lines, such as that they are skew.