Vector Analysis

Table of Contents

Introduction

Vector analysis is a branch of mathematics that deals with quantities that have both magnitude and direction. Unlike scalars, which only have magnitude, vectors are essential in representing physical quantities such as force, velocity, and acceleration.

Vector Basics

A vector is often represented as an arrow with its length proportional to the magnitude and direction pointing towards its intended direction. In a coordinate system, a vector $\vec{a}$ in three dimensions can be expressed as:

$\vec{a} = a_x \hat{i} + a_y \hat{j} + a_z \hat{k}$

where $a_x$ , $a_y$ , and $a_z$ are the components of the vector along the x, y, and z axes, respectively, and $\hat{i}$ , $\hat{j}$ , and $\hat{k}$ are the unit vectors along these axes. Unit vectors have length equal to 1.

Vector Operations

Vector addition and subtraction follow the rule of parallelogram law. If $\vec{a}$ and $\vec{b}$ are two vectors, their addition $\vec{a} + \vec{b}$ and subtraction $\vec{a} - \vec{b}$ are represented as:

$\vec{a} + \vec{b} = (a_x + b_x) \hat{i} + (a_y + b_y) \hat{j} + (a_z + b_z) \hat{k}$

$\vec{a} - \vec{b} = (a_x - b_x) \hat{i} + (a_y - b_y) \hat{j} + (a_z - b_z) \hat{k}$

Scalar multiplication of a vector scales its magnitude without changing its direction:

$c\vec{a} = (ca_x) \hat{i} + (ca_y) \hat{j} + (ca_z) \hat{k}$

where $c$ is a scalar.

The dot product (scalar product) and cross product (vector product) are two essential operations in vector analysis.

The dot product of vectors $\vec{a}$ and $\vec{b}$ is:

$\vec{a} \cdot \vec{b} = a_x b_x + a_y b_y + a_z b_z$

The cross product of vectors $\vec{a}$ and $\vec{b}$ is:

$\vec{a} \times \vec{b} = (a_y b_z - a_z b_y) \hat{i} - (a_x b_z - a_z b_x) \hat{j} + (a_x b_y - a_y b_x) \hat{k}$

Applications

Vector analysis is fundamental in physics and engineering fields, notably in mechanics, electromagnetism, and fluid dynamics. It provides a robust mathematical framework to describe and analyze physical phenomena involving directional quantities.

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Introduction

Vector Basics

Vector Operations

Applications

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Vector Analysis

Introduction

Vector Basics

Vector Operations

Applications

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