Sideway
output.to from Sideway
Draft for Information Only

Content

Vector Components
 Components of Vector
  One Dimension
  Two Dimension
  Three Dimension
 Vector in Space

Vector Components

image A vector is used to represent the magnitude and direction of a vector quantity. A vector quantity, 𝑄, is therefore can be expressed as a simple vector 𝒒. Or in form of a scaled vector by simple geometry operation, that is a scalar 𝑘 times a directional vector 𝒅 or the magnitude |𝒒| times a unit directional vector 𝒅. In other words, the study of vector components can only focus on the directional vector irrespective of its magnitude.

Components of Vector

In general, the values that are used to represent a vector are called the components of the vector, and the number of components used to define a vector is equal to the number of dimensions of interest.

One Dimension

image Since a one-dimensional vector always lies along a line, only one value is needed to specify the vector. There is only one component for a one-dimensional vector. For example, 𝒅=(𝑑)

Two Dimension

image Since a two-dimensional vector always lies in a Cartesian plane, two values are needed to specify the vector. There are two components for a two-dimensional vector. For example, 𝒅=(𝑥,𝑦). image If 𝒅 is a unit vector, the components of the vector can be expressed in terms of angles between the vector and coordinate axes, that is 𝑥=cos 𝛼 and 𝑦=cos 𝛽. Therefore 𝒅=(cos 𝛼,cos 𝛽)
Since the components of a vector is an ordered set, the rectangular Cartesian coordinate system of components 𝑥 and 𝑦 must be arranged in correct order following the right-hand rule.

Three Dimension

image Since a three-dimensional vector always lies in a Cartesian space, three values are needed to specify the vector. There are three components for a three-dimensional vector. For example, 𝒅=(𝑥,𝑦,𝑧). image If 𝒅 is a unit vector, the components of the vector can be expressed in terms of angles between the vector and coordinate axes, that is 𝑥=cos 𝛼, 𝑦=cos 𝛽, and 𝑧=cos 𝛾. Therefore 𝒅=(cos 𝛼,cos 𝛽,cos 𝛾)
Since the components of a vector is an ordered set, the rectangular Cartesian coordinate system of components 𝑥, 𝑦, 𝑧 and must be arranged in correct order following the right-hand rule.

Vector in Space

image Since a point can be considered as a geometric object in a three dimensional space, a point can be represented by the coordinates of a coordinate system such that a vector can be expressed algebraically. The most common system for a three dimensional space is the rectangular coordinate system called Cartesian coordinate system with three mutually perpendicular straight axes of same scale. A vector 𝒂 with initial point 𝐴 and terminal point 𝐵 can be expressed in term of coordinates 𝐴(𝑥1,𝑦1,𝑧1) and 𝐵(𝑥2,𝑦2,𝑧2). The vector 𝒂 can also be interpreted as a displacement vector 𝐴𝐵 displaced from point 𝐴 to point 𝐵. And the geometry of the directed line segment corresponding to the displacement vector can be specified by the numbers 𝑎1=𝑥2-𝑥1, 𝑎2=𝑦2-𝑦1, and 𝑎3=𝑧2-𝑧1 with respect to point 𝐴. These numbers are called components of the vector 𝒂 with respect to the corresponding Cartesian coordinate system because vector 𝒂 can be represented by these components. That is 𝒂=𝐴𝐵=(𝑥2-𝑥1,𝑦2-𝑦1,𝑧2-𝑧1)=(𝑎1,𝑎2,𝑎3) And the length or the magnitude |𝒂| of vector 𝒂 can be determined by the Pythagorean theorem geometrically. image As the components 𝑎1, 𝑎2, and 𝑎3 of vector 𝒂 are derived from the end points of the vector by subtracting the coordinates of initial point from the coordinates of the terminal point, the components of vector are independent of the choice of the initial point of the vector and are dependent on the magnitude and direction of the vector only. In other words, the components of vector 𝒂 is a free vector bounded to point 𝐴 with respect to the corresponding Cartesian coordinate system. image Physically, a free vector can be translated arbitrarily to indicate a vector quantity at a point having equal vector quantity. The terminal point of the vector can be determined uniquely once the initial point of the vector is fixed. If the initial point of a vector is located at the origin, the components of the vector are then equal to the coordinate of the terminal point with respect to the corresponding Cartesian coordinate system. The vector with initial point bounded at the origin is called position vector. Therefore, any vector in three dimensional space can be represented by a position vector by translating the initial point of the vector to the origin. And for a specified Cartesian coordinate system, each vector in space can be mapped to ordered triple components in one-to-one relation with respect to the corresponding Cartesian coordinate system. In other words, two vectors 𝒂 and 𝒃 are equal if and only if the corresponding component of two vectors are equal. That is 𝑎1=𝑏1, 𝑎2=𝑏2, and 𝑎3=𝑏3.
As both the magnitude and the direction of the directed line segment of the corresponding vector can be obtained from the components of the corresponding vector, the geometry of the directed line segment in three dimensional space can also be determined by the ordered triple components of the vector in one-to-one relation with respect to the corresponding Cartesian coordinate system.
However, the vector representation of quantity with magnitude and direction is always dependent on the choice of coordinate system.

©sideway

ID: 191201202 Last Updated: 12/12/2019 Revision: 0 Ref:

close

References

  1. Robert C. Wrede, 2013, Introduction to Vector and Tensor Analysis
  2. Daniel Fleisch, 2012, A Student’s Guide to Vectors and Tensors
  3. Howard Anton, Chris Rorres, 2010, Elementary Linear Algebra: Applications Version
close
IMAGE

Home 5

Business

Management

HBR 3

Information

Recreation

Hobbies 8

Culture

Chinese 1097

English 337

Reference 67

Computer

Hardware 149

Software

Application 198

Digitization 117

Numeric 19

Programming

Web 283

Unicode 494

HTML 65

CSS 58

ASP.NET 92

OS 389

DeskTop 7

Python 19

Knowledge

Mathematics

Formulas 8

Algebra 25

Number Theory 206

Trigonometry 18

Geometry 18

Calculus 67

Complex Analysis 21

Engineering

Tables 8

Mechanical

Mechanics 1

Rigid Bodies

Statics 92

Dynamics 37

Fluid 5

Fluid Kinematics 5

Control

Process Control 1

Acoustics 19

FiniteElement 2

Physics

Electric 27

Biology 1

Geography 1


Copyright © 2000-2020 Sideway . All rights reserved Disclaimers last modified on 06 September 2019