O Level MathematicsC7.5 Vectors (column notation, magnitude from grid, simple addition/subtraction).

🚀 Vector Voyage: Mastering Columns & Magnitudes!

Edudent Academy
26 Jan 26

Vectors pop up all over O-Level Maths, from describing journeys on a map to modeling forces in Physics. **Understanding how to write, add, subtract, and measure vectors quickly is a key skill that can secure easy marks in Paper 2 and Paper 4.**

Column Notation, Direction & Magnitude

In exam questions you will usually see vectors written in column form, e.g. (3−2)\begin{pmatrix} 3 \\ -2 \end{pmatrix}. The top number shows movement in the xx-direction (right is positive, left is negative) and the bottom number shows movement in the yy-direction (up is positive, down is negative). **The magnitude** (length) of a vector is found using Pythagoras:
∥(ab)∥=a2+b2.\left\|\begin{pmatrix} a \\ b \end{pmatrix}\right\| = \sqrt{a^{2}+b^{2}}.
On a square grid, you can literally count the horizontal and vertical steps first, then apply this formula.

  • A vector joining A(1,2)A(1,2) to B(5,7)B(5,7) is (45)\begin{pmatrix} 4 \\ 5 \end{pmatrix}.
  • Magnitude on the grid: 42+52=41 (≈6.40).\sqrt{4^{2}+5^{2}} = \sqrt{41}\,(\approx 6.40).
  • Addition: (2−1)+(−34)=(−13).\begin{pmatrix} 2 \\ -1 \end{pmatrix} + \begin{pmatrix} -3 \\ 4 \end{pmatrix} = \begin{pmatrix} -1 \\ 3 \end{pmatrix}.
  • Subtraction means adding the negative: u−v=u+(−v).\mathbf{u}-\mathbf{v}=\mathbf{u}+(-\mathbf{v}).

Worked Example: Treasure Map Vector

Problem: A pirate walks (62)\begin{pmatrix} 6 \\ 2 \end{pmatrix} steps from point PP to point QQ, then (−45)\begin{pmatrix} -4 \\ 5 \end{pmatrix} steps from QQ to point RR. (i) Find PR→\overrightarrow{PR} in column form. (ii) Calculate its magnitude.

  • Step 1:
    PR→=PQ→+QR→=(62)+(−45)=(27).\overrightarrow{PR}=\overrightarrow{PQ}+\overrightarrow{QR}=\begin{pmatrix} 6 \\ 2 \end{pmatrix}+\begin{pmatrix} -4 \\ 5 \end{pmatrix}=\begin{pmatrix} 2 \\ 7 \end{pmatrix}.
  • Step 2:
    ∣PR→∣=22+72=4+49=53  (≈7.28).|\overrightarrow{PR}|=\sqrt{2^{2}+7^{2}}=\sqrt{4+49}=\sqrt{53}\,\,(\approx 7.28).

Vectors are easy marks once you are fluent with column notation and the Pythagorean shortcut for magnitude. **Practise plotting vectors on grid paper and adding them tip-to-tail**—soon you will breeze through any vector question the examiner can throw at you!