Vectors and Applications
Table of Contents
- Introduction
- Definitions
- Quiz
- Geometric View of Vectors
- Copy of Copy of 1. Cartesian components of vectors
- Vector 3D
- Vector Addition
- Addition & Subtraction
- Vector Addition: The Order Does NOT Matter
- Scalar Multiple
- Scalar Multiples
- Cross & Mixed Product
- Vetor-products
- Vector Applications For Maritime Problems
- What Are the Applications of Vectors?
- Word Problems involving the bearing (direction) of a boat
- Forces as a Vector
- Force Vector Application
- Velocity Vector Applications
- helić
- Vector Application: Ship's Course
Definitions
Definition (vector)
[size=150]A [b]vector[/b] is a directed line segment with an arrowhead at one end. It has an[b][color=#0000ff] initial point A, where it begins, and a terminal point B, where it ends[/color][/b].[br][list][*]It is defined by three components: [color=#980000][i]magnitude, direction and orientation[/i]. [/color][/*][/list][color=#20124d]To every point [/color][b][color=#0000ff]P[/color][/b][color=#20124d] in a plane there can be assigned a unique vector whose initial point is in the origin [/color][b][i]O[/i][/b][color=#20124d] and the terminal point is in the given point P. This vector [/color][math]\vec{OP}[/math][color=#20124d] is called the [b][i]position [/i][/b]or [b][i]radius [/i][/b]vector of the point P.[/color] [br] [br][b][i]Equal[/i][/b] vectors have the same magnitude, the same direction and the same orientation.[br][b]Opposite [/b]vectors have the same direction but the opposite orientation.[br][br][math]\vec{AA}[/math] is called [b][i]zero - vector[/i][/b] and its [u]magnitude is equal to 0. It has the same the initial and terminal point.[/u][/size]
Vector defined with two points and a radius vector
Copy of Copy of 1. Cartesian components of vectors
[color=#9900ff][b][size=150][center]Key Point[/center][/size][/b][/color][size=150]In two dimensions, the [b][color=#980000]unit vectors[/color][/b] in the directions of the two coordinate axes are written as[size=100] [math]\vec{i}[/math] and [math]\vec{j}[/math].[/size][br]If a point P has coordinates (x, y) then the [color=#980000][b]position vector[/b][/color] [math]\vec{\text{OP}}[/math] may be written as a l[u]inear combination of unit vectors: [/u][math]\vec{\text{OP}}=x\vec{i}+y\vec{j}[/math][br]or equivalently as a column vector [math]\vec{OP}=\binom{x}{y}[/math].[br]x and y are [u][color=#0000ff]scalars [/color][/u](real numbers) and they represent the [b][color=#0000ff]coordinates[/color][/b] of the point P.[br]The expression [math]x\vec{i}+y\vec{j}[/math] is a [color=#0000ff][b]vector[/b][/color].[br][br]Given an initial point of [b](0,0)[/b] and terminal point [b](a,b)[/b], a vector may be represented as [color=#0000ff][b](a,b)[/b][/color].[/size]
Addition & Subtraction
Scalar Multiples
Vetor-products
What Are the Applications of Vectors?
What Are the Applications of Vectors?
Almost every branch of engineering today uses vectors as tools. [br][list][*]One of the most common uses of vectors is in solving [u]navigation problems[/u]. These navigation problems use variables such as [color=#0000ff]speed [/color]and [color=#0000ff]direction [/color]to form the computational vector.[/*][*]Some navigation problems require us to find the [b][color=#0000ff]course [/color][/b]of a ship that includes the combined [u]forces of wind and current represented as vectors[/u]. By using vector addition to these different forces, it is possible to create an accurate estimate of the trajectory and distance traveled by the object.[/*][*]Mechanical engineers who design using fluid dynamics concepts use vectors in their calculations to describe real-world forces, such as wind and water motion.[/*][*]Electrical engineers also use them to describe the [u]forces of magnetic and electric fields[/u].[/*][/list] [br][i][color=#980000]For these problems it is important to understand the r[u]esultant of the two forces and the force component.[/u][/color][/i][u][br][/u]