Introduction to Vectors

Overview

Vectors are fundamental mathematical objects that represent quantities with both magnitude and direction. They form the building blocks of linear algebra.

Key Concepts

Concept	Description
Vector	An ordered list of numbers (components)
Column Vector	Vertical arrangement: entries stacked
Row Vector	Horizontal arrangement: entries side by side
Components	Individual entries of a vector
Dimension	Number of components in a vector

Notation

Column Vector

$\mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix} \quad \text{or} \quad \mathbf{v} = (v_1, v_2, \ldots, v_n)^T$

Row Vector

$\mathbf{v} = \begin{bmatrix} v_1 & v_2 & \cdots & v_n \end{bmatrix}$

Geometric Interpretation

In 2D ($\mathbb{R}^2$)

A vector $\mathbf{v} = \begin{bmatrix} a \\ b \end{bmatrix}$ represents:

• A point at coordinates $(a, b)$
• An arrow from origin to $(a, b)$
• A displacement of $a$ units right and $b$ units up

In 3D ($\mathbb{R}^3$)

A vector $\mathbf{v} = \begin{bmatrix} a \\ b \\ c \end{bmatrix}$ represents a point or arrow in 3-dimensional space.

Vector Magnitude (Length)

The magnitude or norm of a vector:

$\|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2}$

Examples

For $\mathbf{v} = \begin{bmatrix} 3 \\ 4 \end{bmatrix}$:

$\|\mathbf{v}\| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

Unit Vectors

A unit vector has magnitude 1. To find the unit vector in the direction of $\mathbf{v}$:

$\hat{\mathbf{u}} = \frac{\mathbf{v}}{\|\mathbf{v}\|}$

Standard Unit Vectors

In $\mathbb{R}^2$:

• $\mathbf{i} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$ (x-direction)
• $\mathbf{j} = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$ (y-direction)

In $\mathbb{R}^3$:

• $\mathbf{i} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$
• $\mathbf{j} = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}$
• $\mathbf{k} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}$

Zero Vector

The zero vector has all components equal to zero:

$\mathbf{0} = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \end{bmatrix}$

Properties:

• $\|\mathbf{0}\| = 0$
• $\mathbf{v} + \mathbf{0} = \mathbf{v}$ for any vector $\mathbf{v}$