Vectors are powerful tools that help us describe and analyze motion, forces, and other physical quantities in multiple dimensions. They're like arrows that show both size and direction, making them perfect for representing things like velocity or force.

In this part, we'll learn how to work with vectors in 2D and 3D space. We'll cover basic operations like adding and subtracting vectors, as well as finding their magnitude and direction. These skills are essential for solving real-world problems in physics and engineering.

Vector Basics and Operations

Vectors in multiple dimensions

Vector definition quantifies magnitude and direction represented by arrow (force, velocity)
Two-dimensional vectors comprise x and y components notated as $\vec{v} = \langle x, y \rangle$ (plane motion)
Three-dimensional vectors include x, y, and z components notated as $\vec{v} = \langle x, y, z \rangle$ (space motion)
Standard basis vectors serve as reference directions
- Two dimensions use $\hat{i}$ and $\hat{j}$ (horizontal, vertical)
- Three dimensions employ $\hat{i}$ , $\hat{j}$ , and $\hat{k}$ (length, width, height)

Vector operations and methods

Vector addition combines vectors tip-to-tail graphically or adds corresponding components algebraically
Vector subtraction reverses direction of subtracted vector graphically or subtracts corresponding components algebraically
Scalar multiplication stretches or shrinks vector graphically or multiplies each component by scalar algebraically
Examples:
1. Add vectors $\vec{a} = \langle 2, 3 \rangle$ and $\vec{b} = \langle -1, 4 \rangle$
2. Subtract $\vec{b}$ from $\vec{a}$
3. Multiply $\vec{a}$ by scalar 2

Vector magnitude and direction

Magnitude calculation measures vector length
- Two dimensions: $|\vec{v}| = \sqrt{x^2 + y^2}$ (Pythagorean theorem)
- Three dimensions: $|\vec{v}| = \sqrt{x^2 + y^2 + z^2}$ (extended Pythagorean theorem)
Direction determination finds angle with positive x-axis
- Use $\theta = \tan^{-1}(\frac{y}{x})$ and adjust for quadrants when necessary
Examples:
1. Calculate magnitude of $\vec{v} = \langle 3, 4 \rangle$
2. Find direction of $\vec{v} = \langle -2, 2 \rangle$

Properties of vectors

Commutative property allows vector addition order change $\vec{a} + \vec{b} = \vec{b} + \vec{a}$ (force combinations)
Associative property permits grouping changes in vector addition $(\vec{a} + \vec{b}) + \vec{c} = \vec{a} + (\vec{b} + \vec{c})$ (multiple force analysis)
Distributive properties relate scalar multiplication and addition
- Scalar multiplication: $c(\vec{a} + \vec{b}) = c\vec{a} + c\vec{b}$ (scaling combined vectors)
- $(c + d)\vec{a} = c\vec{a} + d\vec{a}$ (splitting scalar multiplication)
Zero vector acts as additive identity $\vec{v} + \vec{0} = \vec{v}$ (null displacement)
Negative vector serves as additive inverse $\vec{v} + (-\vec{v}) = \vec{0}$ (canceling forces)