IGCSE Maths Vectors Notes
A vector has magnitude and direction, written as a column vector. Vectors add tip-to-tail and are used in geometric proofs.
Key concepts
- Column vector
- Top number is movement in x, bottom is movement in y.
- Adding vectors
- Add the components; geometrically place them tip-to-tail.
- Scalar multiple
- kv scales the vector; parallel vectors are scalar multiples of each other.
Key formulas
Magnitude
|v| = √(x² + y²)
Parallel test
a = k·b
Vectors are parallel if one is a multiple of the other.
Worked example
Given a = (2, 1) and b = (3, −4), find a + b.
- 1Add the x-components: 2 + 3 = 5.
- 2Add the y-components: 1 + (−4) = −3.
- 3Write the resulting column vector.
Answer: (5, −3)
Common mistakes
✗ Adding x to y components.
✓ Add top-to-top and bottom-to-bottom separately.
✗ Sign errors on subtraction.
✓ Subtracting a vector reverses its direction.
✗ Not showing vectors are parallel with a scalar.
✓ Write one vector as k times the other to prove it.
Quick check
Try these, then reveal the answers.
(1, 2) + (3, 1) = ?Reveal answer
(4, 3)
Magnitude of (3, 4)?Reveal answer
5
Is (2, 4) parallel to (1, 2)?Reveal answer
Yes (×2)
Exam tips
- Keep column-vector layout neat to avoid mixing components.
- In proofs, express paths as sums of known vectors.
- Show a common factor to prove vectors are parallel (and hence points collinear).
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Frequently asked questions
How do I know two vectors are parallel?+
One is a scalar multiple of the other, i.e. a = k·b for some number k.
How do I find the magnitude of a vector?+
Use Pythagoras on its components: |v| = √(x² + y²).
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