Algebra › Vectors
Vectors
A vector has size and direction. It's an arrow you can drag around — and the math of vectors is the math of arrows joined head to tail.
Add two vectors — head to tail
Slide vector b onto the tip of a and you land at a + b.
Components
a = (4, 1)
b = (1, 3)
a + b = (5, 4)
Magnitudes
|a| = 4.12
|b| = 3.16
|a + b| = 6.40
Drag the dots. The faint b arrow after the tip of a shows the head-to-tail rule.
What a vector is
A plain number — like 5 — is a scalar. It has size but no direction. Five degrees, five kilos, five seconds.
A vector has both. "5 km north" is a vector. "A force of 10 N pushing east" is a vector. We write a 2D vector as a column or row of numbers — its components:
v = (3, 4) — three units across, four units up.
Magnitude
The length of a vector is found with Pythagoras:
|v| = √(x² + y²)
So (3, 4) has magnitude √(9 + 16) = 5.
Adding — head to tail
To add two vectors, place them tail-to-head and the sum is the arrow from the start to the end. In components, just add component-wise:
(3, 4) + (1, 2) = (4, 6)
Subtracting is just adding the negative
−v points the opposite way. So a − b is the same as a + (−b) — flip b around and add.Scalar multiplication
Multiplying a vector by a scalar k stretches it by factor |k|. If k is negative, it also flips direction.
2 · (3, 4) = (6, 8)
Where you'll meet vectors
- Velocity in physics — speed plus direction.
- Forces, including gravity and friction.
- Computer graphics — every pixel position is one.
- Navigation — "30 km on a bearing of 045°" is a vector.
Quick check
- Find the magnitude of
(6, 8). - Compute
(2, 5) + (4, −1). - What is
3 · (1, −2)?
Answers: 10, (6, 4), and (3, −6).
Find the magnitude of (6, 8).
Compute (2, 5) + (4, −1).