Combining Vectors
- Addition: To add two vectors, add their components. Geometrically, this is "Tip-to-Tail" addition.
- Scalar Multiplication: Multiplying a vector by a number scales its length but keeps its direction (or reverses it).
Worked Examples
Example 1: Addition
If \(\vec{u} = \langle 1, 2 \rangle\) and \(\vec{v} = \langle 3, -1 \rangle\), find \(\vec{u} + \vec{v}\).
- \(x = 1 + 3 = 4\).
- \(y = 2 - 1 = 1\).
- Result: \(\langle 4, 1 \rangle\).
Example 2: Scaling
Find \(2\vec{u}\).
- \(2 \cdot \langle 1, 2 \rangle = \langle 2, 4 \rangle\).
- The vector is now twice as long.
The Bridge to Quantum Mechanics
The "First Postulate" of Quantum Mechanics is that the sum of any two valid quantum states is also a valid quantum state. This is called the Linearity of the Schrödinger Equation. If a particle can be in State A and it can be in State B, it can also be in a state that is \(0.7 \text{ State A} + 0.3 \text{ State B}\). This "Vector Addition" of states is why electrons can be in a superposition. The math of adding vectors is the math of how the universe builds complexity out of simple parts.