Mathematical Representation of Vectors

Overview

**Representation of Vectors**

» Specify amount or magnitude and direction

→ $3$m east and $4$m north

→ $3$m east and $4$m north-east

» How were the quantities added?

→ $3$m east + $4$m north = $\sqrt{{3}^{2}+{4}^{2}}$ as per Pythagoras theorem

→ $3$m east + $4$m north-east : Making this easy to do is the **objective of "vector algebra"**

» Position of an airplane $ai+bj+ck$

→ 3Okm east (x-axis) = $30i$

→ 40km north (y-axis) = $40j$

→ 4km altitude (z-axis) = 4k

» ** component form** of a vector $ai+bj+ck$

→ $i$,$j$,$k$ represent the direction along $x$, $y$, and $z$ axes

→ $a$, $b$, $c$ represent the amount or magnitude along the given directions

illustrative example

If someone is calculating the position of an airplane, a reasonable specification of the position is

$30$km to the east, $40$km north, and at $4$km altitude above sea level

This specifies - North, East, and Altitude. With these three pieces of information, the position of the airplane is specified clearly.

We have arrived at representing vectors that have magnitude and direction.

The representation is

$30$km East + $40$km North + $4$km Altitude

This can be generalized to a 3D space with three axes x-axis, y-axis, and z-axis. The figure represents x-axis, y-axis, and z-axis. A point 'p' is shown.

The x, y, and z coordinates are given as $(x,y,z)$. The point given in figure is '$(4,7,10)$'.

The point is projected onto x-y plane. That is further projected onto x and y axes.

The airplane can be considered as the point 'p' in the three dimensional space. The representation is simplified as

$4i+7j+10k$

or alternatively

$4\hat{i}+7\hat{j}+10\hat{k}$

Where

$i$ or $\hat{i}$ represents the direction in $x$-axis

$j$ or $\hat{j}$ represents the direction in $y$-axis

$k$ or $\hat{k}$ represents the direction in $z$-axis

Note: $\hat{i}$ is pronounced as i-hat or i-cap.

A 3D vector quantity is represented with **3 components** along the three directions of 3D coordinates.

**Mathematical representation: ** A 3D vector quantity is represented in the form
$ai+bj+ck$

or alternatively

$a\hat{i}+b\hat{j}+c\hat{k}$

Where i, j, k are the directions along x, y, and z axes

and a, b, c are the magnitude along the directions respectively.

examples

Represent OP as a vector.The answer is '$3i+5j$'.

The x-axis component is represented with an $i$ and y-axis component is represented with a $j$.

What is the vector form of $\overline{OP}$? The answer is '$3i+6j+4k$'. Referring to the figure, the component along each of the axes

$3i$: $3$ along x axis

$6j$: $6$ along y axis

$4k$: $4$ along z axis

What is the vector form of OP? The answer is '$-1.3i-\frac{11}{2}j+4k$'.

What is the vector form of OP? $2.4i+\frac{7}{2}j$ or $2.4i+\frac{7}{2}j+0k$ When a two dimensional vector is presented, the component along the third dimension is 0.

A point is located from the x, y, z-axes at distances $2$, $-1.2$, and $1.4$ units respectively. What is the vector representation of the point?

The answer is '$2i-1.2j+1.4k$'

What does a $\overrightarrow{OP}=-.5i+2.1j-.6k$ mean?

the point P is $-0.5$ unit away from origin along x-axis

the point P is $2.1$ unit away from origin along y-axis

the point P is $-0.6$ unit away from origin along z-axis

summary

**Mathematical representation: ** A 3D vector quantity is represented in the form
$ai+bj+ck$

or alternatively

$a\hat{i}+b\hat{j}+c\hat{k}$

Where i, j, k are the directions along x, y, and z axes

and a, b, c are the magnitude along the directions respectively.

Outline

The outline of material to learn vector-algebra is as follows.

Note: Click here for detailed outline of vector-algebra.

• Introduction to Vectors

→ __Introducing Vectors__

→ __Representation of Vectors__

• Basic Properties of Vectors

→ __Magnitude of Vectors__

→ __Types of Vectors__

→ __Properties of Magnitude__

• Vectors & Coordinate Geometry

→ __Vectors & Coordinate Geometry__

→ __Position Vector of a point__

→ __Directional Cosine__

• Role of Direction in Vector Arithmetics

→ __Vector Arithmetics__

→ __Understanding Direction of Vectors__

• Vector Addition

→ __Vector Additin : First Principles__

→ __Vector Addition : Component Form__

→ __Triangular Law__

→ __Parallelogram Law__

• Multiplication of Vector by Scalar

→ __Scalar Multiplication__

→ __Standard Unit Vectors__

→ __Vector as Sum of Vectors__

→ __Vector Component Form__

• Vector Dot Product

→ __Introduction to Vector Multiplication __

→ __Cause-Effect-Relation__

→ __Dot Product : First Principles__

→ __ Dot Product : Projection Form__

→ __ Dot Product : Component Form__

→ __Dot Product With Direction__

• Vector Cross Product

→ __Vector Multiplication : Cross Product __

→ __Cross Product : First Principles__

→ __Cross Product : Area of Parallelogram__

→ __Cross Product : Component Form__

→ __Cross Product : Direction Removed__