How Much This Section Is Tested
The "Geometry and Algebra" module in the official syllabus has two parts: vectors and complex numbers. It tests vector operations and the four operations of complex numbers.
We reviewed 7 past papers containing 333 questions in total, assigning each question to a single knowledge point based on what it mainly tested. Complex numbers and plane vectors account for 6 knowledge points and 19 questions — about 6% of the total.
- Complex numbers: four operations, 6 questions (appearing in 6 of the 7 papers); definition of complex numbers, 3 questions (2 papers)
- Plane vectors: linear operations, 4 questions (4 papers); coordinates and operations, 4 questions (3 papers); concepts, 1 question; dot product, 1 question
The number of questions is small, the calculations are short, and the patterns are consistent. Once you have mastered them, this is a reliable source of marks.
Complex Numbers: Learn These Rules First
Definition
- i² = −1
- A complex number is written z = a + bi, where a is the real part and b is the imaginary part
- If b ≠ 0, z is an imaginary number; if a = 0 and b ≠ 0, z is a pure imaginary number
To decide whether a number is real, imaginary or pure imaginary, you only need to look at its real and imaginary parts. No calculation is required.
Complex Conjugates
The complex conjugate of a + bi is a − bi: the real part stays the same and the imaginary part changes sign.
Four Operations
- Multiplication: (a + bi)(c + di) = (ac − bd) + (ad + bc)i
- Division: multiply both the numerator and the denominator by the complex conjugate of the denominator
The key step in division: whenever the denominator contains an imaginary part, first make the denominator real, then simplify.
Examples
- (1 + i)² = 2i
- (1 + i)/(1 − i) = i
Modulus
|a + bi| = √(a² + b²)
The modulus represents a length, so the result is always a nonnegative real number.
Plane Vectors: Learn These Rules First
Let a = (x₁, y₁) and b = (x₂, y₂).
Linear Operations
- Addition and subtraction: a ± b = (x₁ ± x₂, y₁ ± y₂)
- Scalar multiplication: λa = (λx₁, λy₁)
With coordinates, you simply add, subtract or multiply the corresponding numbers position by position.
Dot Product
a·b = x₁x₂ + y₁y₂ = |a||b|cosθ
Using coordinates is the fastest way to calculate a dot product: multiply the x-coordinates, multiply the y-coordinates, then add the two results.
Modulus
|a| = √(x₁² + y₁²)
Parallel and Perpendicular
- Perpendicular: a ⊥ b ⇔ x₁x₂ + y₁y₂ = 0
- Parallel: a ∥ b ⇔ x₁y₂ − x₂y₁ = 0
Keep these two tests clearly apart: for perpendicularity, check whether the dot product equals 0; for parallelism, check whether x₁y₂ − x₂y₁ equals 0.
Review Advice
- The four operations of complex numbers appeared in 6 of the 7 papers, so this is the content to secure first. Practice these three steps until they are automatic: i² = −1, expanding a product, and multiplying by the conjugate of the denominator.
- For vectors, master coordinate operations first, then understand the geometric meaning of linear operations (the parallelogram rule and the triangle rule). Coordinate operations are easy marks, while the geometric meaning helps you see what a question is really asking.
- Every question in this section involves short calculations, so it makes sense to finish it quickly in the first round of the exam and leave time for harder questions.
For international students, this section has little text and fixed formulas, which makes it a small module that is well suited to building confidence.
Next Steps
Start with the 15-minute diagnostic, then go to the Learning Center and practice these knowledge points one by one until they are familiar.