CSEC MATHEMATICS · MODULE 3 · LESSON C9

Vectors in the Plane: Position Vectors, Magnitude, Direction and Proofs

Position and displacement vectors, magnitude and unit vectors, direction, vector geometry proofs of parallel and collinear lines, and applications to velocity.

Higher Concepts in Secondary Level MathematicsM3 4.1M3 4.2M3 4.3M3 4.4≈ 45 min🎮 3 games
Before you start

By the end of this lesson you can…

1WatchExplainer video 2ReadKey notes & exam tips 3PractisePaper 1 style 4ApplyPaper 2 style
Step 1 · Watch

Explainer video

5 narrated scenes. Press play, listen and read along — the points appear as they are explained. Use ‹ Prev / Next › or the scene bar to jump to any scene.

LESSON C9 · EXPLAINER

Vectors in the Plane: Position Vectors, Magnitude, Direction and Proofs

A boat heads east at 5 km/h while a current pushes north at 2 km/h. Vectors tell you where it really goes — and let you prove geometry facts in three lines.

Narration uses your device’s built-in voice. No sound? Switch to “Captions only” — the captions advance on their own.

Explore in 3D

Vectors in the plane

Change the components and see the sum and the angle.

Step 2 · Read

Key notes

Copy these into your notebook in your own words — that is how they stick.

Position vector

OP = ab for P(a, b).

Displacement

AB = OB − OA (end − start).

Magnitude

|v| = √(x² + y²).

Unit vector

v ÷ |v|; magnitude 1.

Direction

tan θ = y/x from the +x axis; bearing = 90° − θ for a vector in the first quadrant.

Parallel

a = k b ⇒ parallel.

Collinear

AB = k BC ⇒ A, B, C on a straight line (they share B).

Ratio points

T divides PQ in the ratio m : n → OT = (n·p + m·q)/(m + n).

EXAM TIPS & COMMON MISTAKES

  • Always end a proof with a clear statement: “MN = ½PR, so MN is parallel to PR”.
  • AB = OB − OA, not OA − OB: “end minus start”.
  • For a bearing from a vector, draw a quick sketch — the angle from North is NOT the angle from the x-axis.
  • For collinear points mention that the vectors share a common point, otherwise they may only be parallel.
Play & practise

Games

3 quick games on this lesson. Earn XP and badges as you play — your best score for each game is saved. Games are for practice and do not change your exam readiness. More games in the Mathematics Arcade.

Step 3 · Practise

Paper 1 practice — multiple choice

8 items modelled on real CSEC Paper 1 questions. Choose one answer for each, then check. Aim for 80% or better. Work each item on paper first.

CSEC Paper 01 style · SO 4.1

1. The point P has coordinates (−3, 4). Its position vector is

Answer: C. OP = xy = −34.
CSEC Paper 01 style · SO 4.1

2. A is (2, 3) and B is (8, 11). The vector AB is

Answer: D. OB − OA = (6, 8).
CSEC Paper 01 style · SO 4.2

3. The magnitude of the vector 68 is

Answer: B. √(36 + 64) = 10.
CSEC Paper 01 style · SO 4.2

4. A unit vector in the direction of 34 is

Answer: A. Divide by the magnitude 5.
CSEC Paper 01 style · SO 4.3

5. The angle which the vector 68 makes with the positive x-axis is closest to

Answer: B. tan⁻¹(8/6) = 53.1°.
CSEC Paper 01 style · SO 4.4

6. Which statement proves that A, B and C are collinear?

Answer: C. A scalar multiple sharing the point B means the points are on one line.
CSEC Paper 01 style · SO 4.4

7. M and N are the midpoints of OP and OR respectively. MN is

Answer: D. MN = ½(OR − OP) = ½PR.
CSEC Paper 01 style · SO 4.4

8. A boat travels with velocity 52 km/h (east, north). Its speed is

Answer: A. √(25 + 4) = √29 = 5.39.
Step 4 · Apply

Paper 2 practice — structured questions

Write full answers, then reveal the mark scheme and tick the points you earned (17 marks in total). Show all your working — method marks count. Your answers are saved on this device.

CSEC Paper 02 style · Module 3 (Vectors — coordinates)

Question 1 [9 marks]

The points P and Q have coordinates (2, −1) and (6, 2) respectively.

(a) Write PQ as a column vector. [1]

MARK SCHEME — (a) (max 1)

(b) Calculate |PQ|. [2]

MARK SCHEME — (b) (max 2)

(c) Write down the unit vector in the direction of PQ. [2]

MARK SCHEME — (c) (max 2)

(d) Calculate the angle that PQ makes with the positive x-axis, to 1 decimal place. [2]

MARK SCHEME — (d) (max 2)

(e) Find the position vector of the midpoint of PQ. [2]

MARK SCHEME — (e) (max 2)

CSEC Paper 02 style · Module 3 (Vector geometry proof)

Question 2 [8 marks]

In triangle OPQ, OP = u and OQ = v. T is a point on PQ such that PT : TQ = 3 : 2. M and N are the midpoints of OP and OQ respectively.

(a) Express PQ in terms of u and v. [1]

MARK SCHEME — (a) (max 1)

(b) Show that OT = ⅖u + ⅗v. [3]

MARK SCHEME — (b) (max 3)

(c) Show that MN is parallel to PQ and state the ratio MN : PQ. [4]

MARK SCHEME — (c) (max 4)

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Sources

Where this topic appears in CSEC papers

Find these in your own past-paper booklets for extra timed practice. Items above are written in the style of these papers; any adapted from a specific question say so.