Vectors
A vector represents both magnitude (size) and direction, making it fundamentally different from ordinary numbers which only have size. In 2D space, vectors are written as ordered pairs (x, y) or as column notation, where x represents horizontal displacement and y represents vertical displacement. Vector operations follow specific rules that preserve both components simultaneously.
Why it matters
Vectors appear throughout physics and engineering to model forces, velocities, and displacements. In video game development, vectors control character movement and camera angles. GPS navigation systems use vectors to calculate routes and distances between coordinates. Weather forecasting relies on wind velocity vectors to predict storm paths. Computer graphics use vectors for 3D rendering and animation. At GCSE level, vectors form the foundation for A-level mechanics modules and university physics courses. Engineering students encounter vectors in structural analysis, where forces acting on bridges must be calculated using vector addition. Even smartphone accelerometers measure movement using 3D vectors to detect orientation changes.
How to solve vectors
Introduction to Vectors
- A vector has both magnitude (length) and direction.
- Write a 2D vector as (x, y) or as a column.
- Add vectors component by component: (a, b) + (c, d) = (a+c, b+d).
- Scalar multiplication scales both components: k(a, b) = (ka, kb).
Example: (3, 2) + (1, 4) = (4, 6). And 2·(3, 2) = (6, 4).
Worked examples
Write the vector from A(-3, -1) to B(5, 2) as a column vector.
Answer: AB⃗ = (8, 3)
- Subtract coordinates: B − A → (5 − -3, 2 − -1) — Each component of the vector is the difference of the corresponding coordinates.
- Compute → AB⃗ = (8, 3) — x-component: 5 − -3 = 8, y-component: 2 − -1 = 3.
Given a⃗ = (-4, 3) and b⃗ = (4, -3), find a⃗ − b⃗.
Answer: a⃗ − b⃗ = (-8, 6)
- Add/subtract component-wise → (-4 − 4, 3 − -3) — The difference is found by applying the operation to each pair of components.
- Compute → (-8, 6) — x: -4 − 4 = -8, y: 3 − -3 = 6.
Find the length of the vector v⃗ = (0, -1).
Answer: |v⃗| = 1
- Use the magnitude formula: |v⃗| = √(x² + y²) → |v⃗| = √(0² + -1²) — The magnitude is found using the Pythagorean theorem.
- Compute the squares → |v⃗| = √(0 + 1) = √1 — 0² = 0, -1² = 1.
- Evaluate the square root → |v⃗| = 1 — √1 = 1.
Common mistakes
- When finding the vector from A(2, 5) to B(7, 1), writing AB⃗ = (2, 5) instead of correctly calculating AB⃗ = (5, -4) by subtracting A from B.
- Adding vectors incorrectly as (3, 2) + (1, 4) = (3, 6) by only adding the y-components, rather than the correct answer (4, 6).
- Confusing magnitude calculation by writing |(-3, 4)| = -3 + 4 = 1 instead of using the formula √((-3)² + 4²) = 5.