Magnitude and Direction
A scalar is just a number (like temperature). A Vector is a quantity that has both a Size (magnitude) and a Direction (like velocity). We draw them as arrows.
Notation: \(\vec{v} = \langle x, y \rangle\) or \(\vec{v} = x\hat{i} + y\hat{j}\).
Worked Examples
Example 1: Components
A vector has a magnitude of 10 and points at a 30-degree angle. Find its components.
- \(x = 10 \cos(30) = 8.66\).
- \(y = 10 \sin(30) = 5.0\).
- Result: \(\vec{v} = \langle 8.66, 5 \rangle\).
Example 2: Magnitude
Find the magnitude of \(\vec{w} = \langle 3, 4 \rangle\).
- \(|\vec{w}| = \sqrt{3^2 + 4^2} = 5\).
The Bridge to Quantum Mechanics
In Quantum Mechanics, the state of a particle is not a number; it is a Vector in Hilbert Space. While a 2D vector lives on a plane, a quantum state vector lives in an infinite-dimensional space. The "Direction" of the vector tells us the particle's phase, and its "Length" tells us the probability. Just as you can break a 2D vector into \(x\) and \(y\) parts, we break a quantum state into its "basis states" (like energy levels). Vector math is the absolute requirement for the next two years of this course.