Vector Operations

Overview

Vector operations allow us to combine and manipulate vectors. The fundamental operations are addition, scalar multiplication, and products.

Vector Addition

For vectors $\mathbf{u} = \begin{bmatrix} u_1 \\ u_2 \\ \vdots \\ u_n \end{bmatrix}$ and $\mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix}$:

$\mathbf{u} + \mathbf{v} = \begin{bmatrix} u_1 + v_1 \\ u_2 + v_2 \\ \vdots \\ u_n + v_n \end{bmatrix}$

Properties of Addition

Property	Formula
Commutative	$\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}$
Associative	$(\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})$
Identity	$\mathbf{u} + \mathbf{0} = \mathbf{u}$
Inverse	$\mathbf{u} + (-\mathbf{u}) = \mathbf{0}$

Scalar Multiplication

For scalar $c$ and vector $\mathbf{v}$:

$c\mathbf{v} = \begin{bmatrix} cv_1 \\ cv_2 \\ \vdots \\ cv_n \end{bmatrix}$

Properties

Property	Formula
Distributive	$c(\mathbf{u} + \mathbf{v}) = c\mathbf{u} + c\mathbf{v}$
Distributive	$(c + d)\mathbf{v} = c\mathbf{v} + d\mathbf{v}$
Associative	$c(d\mathbf{v}) = (cd)\mathbf{v}$
Identity	$1\mathbf{v} = \mathbf{v}$

Dot Product (Inner Product)

For vectors $\mathbf{u}$ and $\mathbf{v}$ in $\mathbb{R}^n$:

$\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + \cdots + u_nv_n = \sum_{i=1}^{n} u_i v_i$

Properties of Dot Product

• Commutative: $\mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u}$
• Distributive: $\mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = \mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w}$
• Scalar: $(c\mathbf{u}) \cdot \mathbf{v} = c(\mathbf{u} \cdot \mathbf{v})$
• Self: $\mathbf{v} \cdot \mathbf{v} = \|\mathbf{v}\|^2$

Geometric Interpretation

$\mathbf{u} \cdot \mathbf{v} = \|\mathbf{u}\| \|\mathbf{v}\| \cos\theta$

where $\theta$ is the angle between $\mathbf{u}$ and $\mathbf{v}$.

Orthogonality

Two vectors are orthogonal (perpendicular) if:

$\mathbf{u} \cdot \mathbf{v} = 0$

Cross Product (3D Only)

For vectors $\mathbf{u} = \begin{bmatrix} u_1 \\ u_2 \\ u_3 \end{bmatrix}$ and $\mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix}$:

$\mathbf{u} \times \mathbf{v} = \begin{bmatrix} u_2v_3 - u_3v_2 \\ u_3v_1 - u_1v_3 \\ u_1v_2 - u_2v_1 \end{bmatrix}$

Properties of Cross Product

• Anti-commutative: $\mathbf{u} \times \mathbf{v} = -(\mathbf{v} \times \mathbf{u})$
• Distributive: $\mathbf{u} \times (\mathbf{v} + \mathbf{w}) = \mathbf{u} \times \mathbf{v} + \mathbf{u} \times \mathbf{w}$
• Scalar: $c(\mathbf{u} \times \mathbf{v}) = (c\mathbf{u}) \times \mathbf{v} = \mathbf{u} \times (c\mathbf{v})$
• Self: $\mathbf{v} \times \mathbf{v} = \mathbf{0}$

Geometric Interpretation

• $\|\mathbf{u} \times \mathbf{v}\| = \|\mathbf{u}\| \|\mathbf{v}\| \sin\theta$
• Result is perpendicular to both $\mathbf{u}$ and $\mathbf{v}$
• Represents area of parallelogram formed by $\mathbf{u}$ and $\mathbf{v}$

Examples

Example 1: Vector Addition

$\mathbf{u} = \begin{bmatrix} 2 \\ 3 \end{bmatrix}$, $\mathbf{v} = \begin{bmatrix} 1 \\ -2 \end{bmatrix}$

$\mathbf{u} + \mathbf{v} = \begin{bmatrix} 2+1 \\ 3+(-2) \end{bmatrix} = \begin{bmatrix} 3 \\ 1 \end{bmatrix}$

Example 2: Dot Product

$\mathbf{u} = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$, $\mathbf{v} = \begin{bmatrix} 4 \\ -5 \\ 6 \end{bmatrix}$

$\mathbf{u} \cdot \mathbf{v} = 1(4) + 2(-5) + 3(6) = 4 - 10 + 18 = 12$

Example 3: Cross Product

$\mathbf{u} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$, $\mathbf{v} = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}$

$\mathbf{u} \times \mathbf{v} = \begin{bmatrix} 0(0) - 0(1) \\ 0(0) - 1(0) \\ 1(1) - 0(0) \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}$