O-Level E-Maths Coordinate Geometry and Vectors: distance, midpoint and gradient, the equation of a straight line, two-dimensional vectors, and vector geometry with position vectors

What this strand is about

Coordinate geometry and vectors turns geometric questions into algebra. With coordinates you can compute lengths, midpoints and gradients exactly, and with vectors you can describe direction and magnitude and prove geometric facts by calculation rather than measurement. This overview ties the strand together and links to every dot point, each with worked answers and practice.

See the full set of dot points at /sg-o-level/mathematics/syllabus.

Distance, midpoint and gradient

The strand opens with distance, midpoint and gradient. The distance between $(x_1, y_1)$ and $(x_2, y_2)$ is

a direct use of Pythagoras; the midpoint is the average of the coordinates; and the gradient is the change in $y$ over the change in $x$. These three quantities power most coordinate-geometry problems.

The straight line

Coordinate geometry of the straight line finds the equation of a line from a point and a gradient or from two points, and uses the gradient conditions: parallel lines have equal gradients, and perpendicular lines have gradients whose product is $-1$ (each is the negative reciprocal of the other). These conditions let you build a line related to a given one.

Vectors in two dimensions

Vectors in two dimensions introduces the column vector $\begin{pmatrix} a \\ b \end{pmatrix}$, addition and subtraction (component by component), scalar multiplication, and the magnitude $\sqrt{a^2 + b^2}$. A vector carries both direction and size, which is what makes it more than a pair of numbers.

Vector geometry and position vectors

Vector geometry and position vectors is the proof-oriented finish. Position vectors locate points relative to the origin, and the route method expresses one vector via others ($\vec{AB} = \vec{OB} - \vec{OA}$), letting you prove that lines are parallel, that points are collinear, and to find the ratio in which a point divides a segment.

How the strand is examined

• Keep the order of subtraction consistent. In the distance and gradient formulas, subtract the coordinates in the same order for both $x$ and $y$.
• Use the right gradient condition. Equal gradients for parallel; product $-1$ (negative reciprocal) for perpendicular.
• Build vectors by a clear route. In vector geometry, write $\vec{AB} = \vec{OB} - \vec{OA}$ and state the parallel or ratio conclusion explicitly.

Check your knowledge

Attempt these, then check the solutions.

1. Find the distance between $(1, 2)$ and $(4, 6)$. (2 marks)
2. Find the midpoint of $(-3, 5)$ and $(7, 1)$. (2 marks)
3. State the gradient of a line perpendicular to one with gradient $3$. (1 mark)
4. Given $\mathbf{a} = \begin{pmatrix} 2 \\ -1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$, find $\mathbf{a} + \mathbf{b}$. (2 marks)
5. Find the magnitude of the vector $\begin{pmatrix} 6 \\ 8 \end{pmatrix}$. (2 marks)