Ch 3 · 13 Vectors in Two Dimensions · O Level Additional Mathematics 4037
Topic 3 of 3 in Series, Vectors and Counting — O Level Additional Mathematics 4037 — 57 lessons.
Introduction
A boat crossing a river is pushed sideways by the current; a drone's position after 10 seconds depends on where it started and its velocity. Vectors handle both size and direction, so they are the natural language for movement. This topic works in two dimensions. (Syllabus topic 13.)
Key ideas — Notation
A vector can be written as beginpmatrix 3 \\ 4 endpmatrix, as 3i + 4j (where i and j are unit vectors along the x- and y-axes), as overrightarrowAB, or as a bold letter a (underline it in handwriting). Use correct notation in answers.
🧑🏫 Teacher's explanation
Imagine you are giving someone directions in a city laid out in a perfect grid. If you tell them to 'walk 3 blocks East and 4 blocks North', you have given them a vector. You haven't told them where to start, just how to move. In mathematics, a vector represents this 'instruction' of movement rather than a fixed point.
📖 Definitions
🔍 Worked example — Easy
Given a = 3i - 2j and b = beginpmatrix -1 \\ 5 endpmatrix, find a + b in component form.
🔍 Worked example — Standard
Find the unit vector in the direction of v = beginpmatrix 8 \\ -6 endpmatrix.
🔍 Worked example — Exam-style
Given overrightarrowOP = 2i + 3j and overrightarrowOQ = 5i - j, find the vector overrightarrowPQ.
⚠️ Common errors
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🎯 Exam question
Key ideas — Magnitude and unit vectors
|xi + yj| = √(x^2 + y^2)
🧑🏫 Teacher's explanation
Think of a vector as a set of instructions for a journey. Instead of saying 'walk 5km North and 3km East', a vector gives you the exact displacement from a starting point. The magnitude is simply the total distance of the straight-line path from the start to the finish. If you imagine walking along the sides of a right-angled triangle, the magnitude is the length of the hypotenuse.
📖 Definitions
🔍 Worked example — Easy
Find the magnitude of the vector v = 5i - 12j.
🔍 Worked example — Standard
Given a = 3i + 4j, find the unit vector mathbfhata in the direction of a.
🔍 Worked example — Exam-style
A particle moves from point A(2, -3) to point B(5, 1). Find the unit vector in the direction overrightarrowAB.
⚠️ Common errors
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🎯 Exam question
Key ideas — Position vectors and geometry
The position vector of A is overrightarrowOA = a. Then overrightarrowAB = b - a.
🧑🏫 Teacher's explanation
Think of a vector as a set of instructions for moving from one place to another. If you are standing at point O (the origin) and want to get to point A, your instruction is the position vector a. If you then want to go from A to B, you aren't starting from the origin anymore; you are starting from A. To find this 'displacement' vector overrightarrowAB, you take your final destination b and subtract your starting point a, resulting in b - a.
📖 Definitions
🔍 Worked example — Easy
Given position vectors overrightarrowOA = beginpmatrix 2 \\ 3 endpmatrix and overrightarrowOB = beginpmatrix 5 \\ -1 endpmatrix, find the vector overrightarrowAB.
🔍 Worked example — Standard
Point M is the midpoint of line segment PQ. If overrightarrowOP = 3i + 4j and overrightarrowOQ = 7i - 2j, find the position vector overrightarrowOM.
🔍 Worked example — Exam-style
Given overrightarrowAB = kbeginpmatrix 2 \\ -3 endpmatrix and overrightarrowBC = beginpmatrix 6 \\ -9 endpmatrix. Find the value of k and state the relationship between points A, B, and C.
⚠️ Common errors
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🎯 Exam question
Key ideas — Equating like vectors
If pi + qj = ri + sj, then p = r and q = s. In geometry problems, express the same vector in two ways (e.g. along two routes) and equate the coefficients of two non-parallel vectors to find unknown scalars.
🧑🏫 Teacher's explanation
Imagine you are giving someone directions in a city laid out in a perfect grid. If you tell them to go 3 blocks East and 4 blocks North, that is exactly the same destination as telling them to go 3 blocks East and 4 blocks North. Even if you take a different path, as long as your total displacement in the 'East' direction and 'North' direction is the same, you have described the same vector.
📖 Definitions
🔍 Worked example — Easy
Given that 3i + kj = 3i + 7j, find the value of k.
🔍 Worked example — Standard
Given a = beginpmatrix 2 endpmatrix and b = beginpmatrix x \\ 5 endpmatrix, find x if 2a + b = beginpmatrix 7 \\ 11 endpmatrix.
🔍 Worked example — Exam-style
In a triangle OAB, vecOA = 3i + 2j and vecOB = 5i + 6j. If vecAB = c, find c in terms of i and j.
⚠️ Common errors
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🎯 Exam question
Key ideas — Velocity and position
A particle starting at position vector r_0 and moving with constant velocity v has position after time t:
🧑🏫 Teacher's explanation
Imagine you are walking across a moving escalator. Your position on the floor depends on two things: where you started (your initial position) and how much you have moved due to your own walking plus the escalator's movement (your velocity multiplied by time). In vector terms, we treat these as 'instructions' for movement.
📖 Definitions
🔍 Worked example — Standard
A boat moves with velocity v_b = 5i + 2j relative to the water. The river current has a velocity v_c = -2i + 3j. Find the resultant velocity v_R of the boat relative to the shore.
🔍 Worked example — Exam-style
Particle A has position r_A = beginpmatrix 1 \\ 2 endpmatrix + tbeginpmatrix 3 \\ 4 endpmatrix and Particle B has position r_B = beginpmatrix 10 \\ 14 endpmatrix + tbeginpmatrix 1 \\ 2 endpmatrix. Find the time t when the particles collide.
⚠️ Common errors
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🎯 Exam question
Worked examples
Find λ and μ such that λ(2i + j) + μ(i - 3j) = 7i - 7j. i: 2λ + μ = 7; j: λ - 3μ = -7. So λ = 2, μ = 3.
🔍 Worked example — Easy
Find the values of λ and μ such that λ(2i + j) + μ(i - 3j) = 7i - 7j.
🔍 Worked example — Standard
A boat heads due north at 5 m s^-1 in a river flowing due east at 2 m s^-1. Find the magnitude and bearing of the resultant velocity.
🔍 Worked example — Exam-style
At t = 0, particle P is at i + 5j with velocity 2i + j, and Q is at 7i + 3j with velocity -i + 2j. Determine if they collide and at what time and position.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) 13; 1/13(5i - 12j). (2) overrightarrowAB = 6i - 8j, |overrightarrowAB| = 10; midpoint 5i - j. (3) speed 5 m s⁻¹; r = 19i + 10j m. (4) overrightarrowOP = a + tfrac34(b - a) = tfrac14a + tfrac34b.