Ch 7 · 3.7 Vectors · A Level Mathematics 9709
Topic 7 of 9 in Pure Mathematics 2 and 3 — A Level Mathematics 9709 — 82 lessons.
Introduction
A drone's flight path, the edges of a pyramid, the line of a laser beam: in three dimensions, coordinate geometry with y = mx + c no longer works, but vectors do. This Paper 3 topic finds lengths, angles, equations of lines and where (or whether) two lines meet. (Syllabus section 3.7, Paper 3 only.)
Key ideas — Notation and basic operations
A vector has size and direction. In 3D, beginpmatrix 3 \\ -2 \\ 2 endpmatrix = 3i - 2j + 2k, where i, j, k are unit vectors along the axes. Add or subtract component by component; multiplying by a scalar k multiplies each component and gives a parallel vector.
🧑🏫 Teacher's explanation
Think of a vector not as a single number, but as a set of instructions for movement. If I tell you to walk 3 meters East and 2 meters North, you aren't at a single point; you are following a specific path. In 3D, we just add a third instruction: 'up' or 'down'.
📖 Definitions
🔍 Worked example — Easy
Given position vectors a = 3i - 2j and b = 5i + 4j, find the displacement vector overrightarrowAB and its magnitude |overrightarrowAB|.
🔍 Worked example — Standard
The position vectors of points A and B are a = beginpmatrix 1 \\ 2 \\ -3 endpmatrix and b = beginpmatrix 4 \\ 6 \\ 1 endpmatrix. Find the position vector of the midpoint of AB.
🔍 Worked example — Exam-style
Find the angle between vectors a = 2i + j - k and b = i + 3j + 2k using the scalar product.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Magnitude and unit vectors
|beginpmatrix x \\ y \\ z endpmatrix | = √(x^2 + y^2 + z^2)
🧑🏫 Teacher's explanation
Think of a vector as a set of instructions for moving from one place to another. If a vector a tells you to walk 3 meters East and 4 meters North, its magnitude is the actual straight-line distance you traveled from your starting point. We use Pythagoras' theorem in 3D to find this distance: √(x^2 + y^2 + z^2).
📖 Definitions
🔍 Worked example — Easy
Find the magnitude of the vector v = 3i - 4j.
🔍 Worked example — Standard
Given position vectors a = beginpmatrix 1 \\ 2 \\ 3 endpmatrix and b = beginpmatrix 4 \\ 6 \\ 3 endpmatrix, find the unit vector in the direction of overrightarrowAB.
🔍 Worked example — Exam-style
A point P has position vector p = 2i + j - 3k. Find the unit vector perpendicular to the vector d = i + j that lies in the xy-plane.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — The scalar product
acdotb = a_1b_1 + a_2b_2 + a_3b_3 = |a||b|cosθ
🧑🏫 Teacher's explanation
Imagine you are walking on a slanted hill. The scalar product tells you how much of your movement is actually going 'forward' in a specific direction, rather than just going up or down. It combines the lengths of your paths with the cosine of the angle between them to give a single number.
📖 Definitions
🔍 Worked example — Easy
Given vectors a = beginpmatrix 3 \\ -2 \\ 1 endpmatrix and b = beginpmatrix 4 \\ 2 \\ -5 endpmatrix, calculate acdotb.
🔍 Worked example — Standard
Find the acute angle θ between the vectors u = 2i + j and v = i + 3j.
🔍 Worked example — Exam-style
A point P has position vector p = beginpmatrix 1 \\ 2 \\ 3 endpmatrix. A line L has direction vector d = beginpmatrix 2 \\ -1 \\ 4 endpmatrix. Find the foot of the perpendicular from P to the line L, assuming the line passes through the origin O.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Vector equation of a line
r = a + tb
🧑🏫 Teacher's explanation
Think of a vector equation of a line as a set of instructions for a traveler. If you want to describe a path, you first need a starting point (a position vector) and a direction to walk in (a direction vector).
📖 Definitions
🔍 Worked example — Easy
Find the vector equation of a line that passes through the point A with position vector a = beginpmatrix 2 \\ -3 \\ 5 endpmatrix and is parallel to the vector b = beginpmatrix 1 \\ 4 \\ -2 endpmatrix.
🔍 Worked example — Standard
Find the vector equation of a line passing through points A(1, 2, 3) and B(4, 0, 8).
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Two lines: parallel, intersecting or skew
🧑🏫 Teacher's explanation
Imagine two laser beams in a dark room. If they are aimed in exactly the same direction, they are parallel. If they are aimed differently but happen to cross at a single point, they are intersecting. However, in 3D space, there is a third possibility: the beams might be aimed in different directions and pass each other at different depths without ever touching. These are called skew lines.
📖 Definitions
🔍 Worked example — Easy
Consider two lines in 3D with equations: L_1: r = beginpmatrix 1 \\ 2 \\ 3 endpmatrix + sbeginpmatrix 2 \\ 4 \\ 6 endpmatrix L_2: r = beginpmatrix 3 \\ 6 \\ 9 endpmatrix + tbeginpmatrix 1 \\ 2 \\ 3 endpmatrix Determine the relationship between L_1 and L_2.
🔍 Worked example — Standard
Line L_1 has equation r = beginpmatrix 1 \\ 0 \\ 2 endpmatrix + sbeginpmatrix 2 \\ 1 \\ -1 endpmatrix and Line L_2 has equation r = beginpmatrix 3 \\ 2 \\ 0 \\ endpmatrix + tbeginpmatrix 1 \\ 1 \\ 1 endpmatrix. Determine if they are parallel, intersecting, or skew.
🔍 Worked example — Exam-style
Find the point of intersection of the lines L_1: r = beginpmatrix 2 \\ 1 \\ 3 endpmatrix + sbeginpmatrix 1 \\ -2 \\ 1 endpmatrix and L_2: r = beginpmatrix 1 \\ 3 \\ 1 endpmatrix + tbeginpmatrix 2 \\ 1 \\ 0 endpmatrix.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
(a) Show that L_1 and L_2 are not parallel. [2] (b) Determine whether the lines intersect or are skew. [3] [5 marks]
Key ideas — Foot of the perpendicular
To find the foot F of the perpendicular from a point P to the line r = a + tb: write F as a general point a + tb, form overrightarrowPF, set overrightarrowPFcdotb = 0 and solve for t. The perpendicular distance from P to the line is |overrightarrowPF|. (The vector product and the distance between skew lines are not required.)
🧑🏫 Teacher's explanation
Imagine you are standing at a point P and you want to walk to a straight road (the line) in the shortest possible way. The shortest path is always a straight line that hits the road at a right angle. The spot where you hit the road is called the 'foot of the perpendicular', which we label F.
📖 Definitions
🔍 Worked example — Easy
Find the foot of the perpendicular from point P(1, 5) to the line with equation r = beginpmatrix 4 \\ 2 \\ 1 endpmatrix + tbeginpmatrix 1 \\ -1 \\ 2 endpmatrix.
🔍 Worked example — Standard
A point P has position vector p = 2i + 3j. A line has equation r = (i + 2j) + t(3i - j). Find the coordinates of the foot of the perpendicular from P to the line.
🔍 Worked example — Exam-style
The line L has equation r = beginpmatrix 1 \\ 0 \\ 2 endpmatrix + tbeginpmatrix 2 \\ 1 \\ -2 endpmatrix. Find the shortest distance from point P(3, 3, 3) to the line L.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Worked examples — Example 1: line through two points
A(1, 2, 3), B(4, 0, 5). overrightarrowAB = beginpmatrix 3 \\ -2 \\ 2 endpmatrix, AB = √(17). Line AB: r = beginpmatrix 1 \\ 2 \\ 3 endpmatrix + sbeginpmatrix 3 \\ -2 \\ 2 endpmatrix.
🔍 Worked example — Easy
Given points A(1, 2, 3) and B(4, 0, 5), find the displacement vector overrightarrowAB and its magnitude |overrightarrowAB|.
🔍 Worked example — Standard
Find the vector equation of the line passing through A(1, 2, 3) and B(4, 0, 5).
🔍 Worked example — Exam-style
A line has the equation r = beginpmatrix 5 \\ -1 \\ 2 endpmatrix + λ beginpmatrix 2 \\ 4 \\ -2 endpmatrix. Find the coordinates of the point on the line where the y-coordinate is 7.
⚠️ Common errors
Worked examples — Example 2: intersecting lines and the angle between them
l_2: r = beginpmatrix 2 \\ -2 \\ 7 endpmatrix + tbeginpmatrix 1 \\ 1 \\ -1 endpmatrix. Does it meet line AB?
🔍 Worked example — Easy
Line L_1 has equation r = beginpmatrix 1 \ 2 \ 3 endpmatrix + sbeginpmatrix 1 \ 0 \ 2 endpmatrix. Line L_2 has equation r = beginpmatrix 4 \ 2 \ 9 endpmatrix + tbeginpmatrix 1 \ 1 \ 1 endpmatrix. Show that the lines intersect and find the point of intersection.
🔍 Worked example — Standard
Find the acute angle between the lines r = beginpmatrix 1 \ 1 \ 1 endpmatrix + tbeginpmatrix 2 \ -1 \ 3 endpmatrix and r = beginpmatrix 0 \ 0 \ 0 endpmatrix + sbeginpmatrix 1 \ 1 \ 1 endpmatrix.
🔍 Worked example — Exam-style
Line L_1 is given by r = beginpmatrix 2 \ -2 \ 7 endpmatrix + tbeginpmatrix 1 \ 1 \ -1 endpmatrix and Line L_2 is given by r = beginpmatrix 3 \ -4 \ 5 \ endpmatrix + sbeginpmatrix 1 \ 1 \ 1 \ endpmatrix. Determine if they intersect, and if so, find the intersection point and the acute angle between them.
⚠️ Common errors
Worked examples — Example 3: skew lines
l_3: r = beginpmatrix 0 \\ 1 \\ 0 endpmatrix + ubeginpmatrix 2 \\ 1 \\ 4 endpmatrix. Not parallel to AB. 1 + 3s = 2u and 3 + 2s = 4u give s = tfrac14, u = tfrac78. Check the y-equation: 2 - 2s = 1.5 but 1 + u = 1.875. Not equal, so the lines are skew.
🔍 Worked example — Easy
Determine if the lines L_1: r = beginpmatrix 1 \ 2 \ 3 endpmatrix + λ beginpmatrix 1 \ 0 \ 1 endpmatrix and L_2: r = beginpmatrix 4 \ 5 \ 6 endpmatrix + μ beginpmatrix 0 \ 1 \ 1 endpmatrix are parallel, intersecting, or skew.
🔍 Worked example — Standard
Show that the lines L_1: r = beginpmatrix 0 \ 1 \ 0 pmatrix + ubeginpmatrix 2 \ 1 \ 4 pmatrix and L_2: r = beginpmatrix 1 \ 0 \ 1 pmatrix + sbeginpmatrix 1 \ 3 \ 2 pmatrix are skew.
🔍 Worked example — Exam-style
Line L_1 has equation r = beginpmatrix 2 \ 3 \ 1 pmatrix + tbeginpmatrix 1 \ 2 \ -1 pmatrix. Line L_2 has equation r = beginpmatrix 1 \ 1 \ 1 pmatrix + sbeginpmatrix 2 \ 1 \ 1 pmatrix. Find the intersection point if it exists, or state if they are skew.
⚠️ Common errors
Worked examples — Example 4: foot of the perpendicular
Find the foot of the perpendicular from P(6, 3, 5) to line AB, and the distance.
🔍 Worked example — Easy
Find the foot of the perpendicular from point P(1, 2, 3) to the line with position vector r = beginpmatrix 2 \\ 0 \\ 1 endpmatrix + tbeginpmatrix 1 \\ 1 \\ 1 endpmatrix.
🔍 Worked example — Standard
A point P has coordinates (6, 3, 5). A line L has the equation r = beginpmatrix 1 \\ 2 \\ 3 endpmatrix + sbeginpmatrix 3 \\ -2 \\ 2 endpmatrix. Find the foot of the perpendicular F from P to L.
🔍 Worked example — Exam-style
Find the shortest distance from point P(6, 3, 5) to the line r = beginpmatrix 1 \\ 2 \\ 3 endpmatrix + sbeginpmatrix 3 \\ -2 \\ 2 endpmatrix.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) tfrac13(2i - j + 2k). (2) cosθ = 8 - 6/3 × 5 = 2/15, θ = 82.3^°. (3) overrightarrowOB = beginpmatrix 3 \\ 5 \\ 2 endpmatrix; M = tfrac12beginpmatrix 3 \\ 5 \\ 2 endpmatrix = (1.5, 2.5, 1) (also the midpoint of OB). (4) 2 - p + 12 = 0 ⇒ p = 14. (5) 1 + t = 4 ⇒ t = 3; y: 6 = 5 + s ⇒ s = 1; check z: 2 - 3 = -1 = 0 - 1 ✓. They meet at 4i + 6j - k.