H2 Maths Basic Vector Properties | Free Notes

Q: What does H2 Maths Notes (JC 1-2): 3.1) Basic Properties of Vectors cover?
A: Vector notation, magnitude-direction forms, and line representations for H2 Maths Topic 3.1.

Before you revise
Keep vector diagrams clean-label initial points, direction arrows, and magnitudes. Many marks are lost to orientation errors or forgetting column-vector format.

• A vector has size and direction: Write components clearly.
• Position vectors locate points; direction vectors describe movement: Decide which one the question uses.
• Lines need a point and a direction: Form $\vec{r}=\vec{a}+\lambda\vec{d}$

Concrete example: If $A(1,2,-3)$ and $B(4,-1,5)$, the direction from A to B is $\vec{OB}-\vec{OA}$

Status: SEAB's current H2 Mathematics (9758) syllabus PDF is labelled for 2026. Topic 3.1 focuses on vector notation, basic operations, and vector equations of lines.

Core Definitions

• A vector $\vec{v}$ has magnitude $\lVert \vec{v} \rVert$

Vector Operations

• Addition/subtraction component-wise: $\vec{a} \pm \vec{b} = \begin{pmatrix} a_1 \pm b_1 \ a_2 \pm b_2 \ a_3 \pm b_3 \end{pmatrix}$

Example -- Midpoint

If $A(2, -1, 5)$ and $B(-4, 7, 1)$, then

$$\vec{OM} = \frac{1}{2}\begin{pmatrix} 2 + (-4) \ -1 + 7 \ 5 + 1 \end{pmatrix} = \begin{pmatrix} -1 \ 3 \ 3 \end{pmatrix}.$$

Lines in Vector Form

• A line through point $A$ with direction vector $\vec{d}$ is $\vec{r} = \vec{OA} + \lambda \vec{d}$

Before writing the line equation, separate the three vector roles:

So a line through two points should use one position vector for the anchor and a displacement vector for direction. Do not use $\vec{OB}$ as the direction just because point $B$ lies on the line.

Example -- Line through two points

Given $A(1, 2, -3)$ and $B(4, -1, 5)$, $\vec{d} = \begin{pmatrix} 3 \ -3 \ 8 \end{pmatrix}$

Angles Between Vectors

• Use dot product: $\vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3$

Example -- Angle calculation

Let $\vec{a} = \begin{pmatrix} 2 \ -1 \ 3 \end{pmatrix}$ and $\vec{b} = \begin{pmatrix} 1 \ 4 \ -2 \end{pmatrix}$

$$\vec{a} \cdot \vec{b} = 2 \times 1 + (-1) \times 4 + 3 \times (-2) = -8.$$

$$\lVert \vec{a} \rVert = \sqrt{14}, \quad \lVert \vec{b} \rVert = \sqrt{21}, \quad \cos\theta = \frac{-8}{\sqrt{14} \sqrt{21}}.$$

Planar Geometry Basics

• Coplanar points share the same plane; check by showing direction vectors are linearly dependent.
• Area of triangle from two vectors $\vec{u}, \vec{v}$: $\tfrac{1}{2} \lVert \vec{u} \times \vec{v} \rVert$

Coplanarity checkpoint

For four points, choose one anchor point first. Then test whether the third displacement can be built from the first two displacement vectors.

Worked check: let $\vec{AB} = \begin{pmatrix}2 \ 1 \ 2\end{pmatrix}$, $\vec{AC} = \begin{pmatrix}1 \ -1 \ 3\end{pmatrix}$

$$4 = 2s + t, \quad 0 = s - t, \quad 5 = 2s + 3t.$$

The second equation gives $s = t$. The first then gives $s = t = \dfrac{4}{3}$, but the third would require $5 = \dfrac{20}{3}$

Common trap: do not test each vector against the origin separately. Coplanarity is about whether the displacement vectors from the same anchor fit into one plane.

Calculator Workflow

• Store vectors in GC matrices or vector memory for quick dot products and norms.
• Use RREF to test linear dependence and solve vector equations.
• Document cross product commands if your GC supports them; otherwise show manual determinant evaluation.

Exam Watch Points

• Express final answers as exact radicals unless context demands decimals.
• Always quote the parameter range when interpreting line equations (e.g. segment $0 \leq \lambda \leq 1$).
• Draw diagrams for vector proofs-markers award method marks for clear geometry.
• Distinguish between position vectors and direction vectors to avoid mixing origins.

Practice Quiz

Consolidate magnitude, direction, and coplanarity checks before tackling product operations.

Quick Revision Checklist

• Convert between vector, parametric, and component descriptions fluently.
• Calculate magnitudes, unit vectors, and angles with precise working.
• Determine lines through points and analyse parallelism/perpendicularity correctly.
• Check collinearity/coplanarity by showing vectors are linearly dependent (or by solving parameter systems).

Next steps: Stay on the H2 Maths notes hub and continue with Topic 3.2 - Scalar & vector products for projection and geometry drills.

Want weekly guided practice on Basic Vector Properties? Our H2 Maths tuition programme builds fluency in this topic through structured problem sets and exam-style drills.

Common exam mistakes

• Confusing position vectors with direction vectors: A position vector $\vec{OA}$ points from the origin to a specific point; a direction vector describes orientation along a line with no fixed starting point. Using a position vector as a direction vector (or vice versa) produces a wrong line equation.
• Forgetting the parameter range when describing a line segment: The vector equation $\vec{r} = \vec{OA} + \lambda\vec{d}$

Frequently asked questions

Are vectors in Paper 1 or Paper 2?
Topic 3 (Vectors) is Pure Mathematics and can appear in Paper 1 (100 marks) or Paper 2 Section A (40 marks). Vector geometry questions are often long-structured questions worth 8–12 marks.

What is the difference between collinearity and coplanarity?
Three points are collinear if they all lie on the same line. Four points are coplanar if they all lie in the same plane. To test collinearity, show direction vectors are parallel with a shared point. To test coplanarity, find a plane equation and verify all four points satisfy it.

Do I need to draw a 3D diagram in the exam?
Yes. A clear labelled diagram earns method marks and helps you interpret vector relationships correctly. Even a rough 3D sketch showing the line direction, a point, and key vectors is sufficient - precision is less important than clarity.

Sources

• SEAB H2 Mathematics syllabus (9758), examinations from 2026 - Topic 3.1 Vectors (basic properties, vector equations of lines, angles between vectors, collinearity/coplanarity via linear dependence) under Section A Pure Mathematics: SEAB H2 Mathematics syllabus PDF