Concept of a Vector
Big Idea:
A vector is a mathematical object that has both magnitude (size/length) and direction. Unlike ordinary numbers (scalars), vectors tell us not just "how much" but also "which way."
Scalar vs. Vector
| Scalar | Vector |
|---|---|
| Temperature: 25°C | Wind: 30 km/h North |
| Mass: 5 kg | Force: 10 N at 45° |
| Speed: 60 km/h | Velocity: 60 km/h East |
| Only magnitude | Magnitude + Direction |
Formal Definition
A vector is a directed line segment. If a vector goes from point $A$ to point $B$, we write it as $\vec{AB}$.
- Initial point (tail): $A$ — where the vector starts
- Terminal point (head): $B$ — where the vector ends (the arrow tip)
- Magnitude: $|\vec{AB}|$ — the length of the segment
- Direction: the angle the vector makes (often measured from the positive $x$-axis)
We also denote vectors as $\vec{a}$, $\vec{b}$, or in bold: a, b.