Vectors
A vector in two dimensions represents both magnitude (length) and direction, written as an ordered pair (x, y) where x and y are the horizontal and vertical components. Unlike a scalar quantity such as temperature or speed, a vector contains directional information that makes it essential for describing motion, forces, and position changes. The vector from point A(-3, 1) to point B(4, 6) is written as (7, 5), found by subtracting the starting coordinates from the ending coordinates.
Why it matters
Vectors form the foundation for physics concepts like velocity, acceleration, and force, where both magnitude and direction matter. In computer graphics, vectors control object movement and rotation, with game engines processing thousands of 2D vector calculations per second. Navigation systems use vectors to represent displacement — a GPS might calculate that traveling 3 miles east and 4 miles north results in a displacement vector of (3, 4) with magnitude 5 miles. Engineering applications include structural analysis, where forces acting on bridges are represented as vectors. The dot product operation determines whether two forces are perpendicular, crucial for stability calculations. Advanced mathematics builds on 2D vectors to introduce 3D space, vector calculus, and linear transformations that appear in machine learning algorithms.
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(4, 6) as a column vector.
Answer: AB⃗ = (7, 5)
- Subtract coordinates: B − A → (4 − -3, 6 − 1) — Each component of the vector is the difference of the corresponding coordinates.
- Compute → AB⃗ = (7, 5) — x-component: 4 − -3 = 7, y-component: 6 − 1 = 5.
Given a⃗ = (4, -2) and b⃗ = (7, 6), find a⃗ − b⃗.
Answer: a⃗ − b⃗ = (-3, -8)
- Add/subtract component-wise → (4 − 7, -2 − 6) — The difference is found by applying the operation to each pair of components.
- Compute → (-3, -8) — x: 4 − 7 = -3, y: -2 − 6 = -8.
Find the length of the vector v⃗ = (-1, 1).
Answer: |v⃗| = √2 ≈ 1.41
- Use the magnitude formula: |v⃗| = √(x² + y²) → |v⃗| = √(-1² + 1²) — The magnitude is found using the Pythagorean theorem.
- Compute the squares → |v⃗| = √(1 + 1) = √2 — -1² = 1, 1² = 1.
- Evaluate the square root → |v⃗| = √2 ≈ 1.41 — √2 = √2 ≈ 1.41.
Common mistakes
- A common error is writing the vector from A(2, 3) to B(5, 1) as (2, 3) instead of (3, -2), forgetting that vectors represent displacement, not position.
- When adding vectors (4, -1) + (2, 5), some write (6, 4) but forget the negative sign, incorrectly getting (6, 6) instead of the correct (6, 4).
- For magnitude calculations, some compute |(-3, 4)| = √(-3 + 4) = 1 instead of using the correct formula √((-3)² + 4²) = √25 = 5.
- In scalar multiplication, 3 · (2, -4) sometimes becomes (6, -4) instead of (6, -12), with the scalar applied to only one component.