Difference between revisions of "Applications-of-GeoGebra/C2/Vectors-and-Matrices/English"

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'''Vector'''  
 
'''Vector'''  
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|| A '''vector''' is a quantity that has both '''magnitude''' and '''direction'''.  
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|| '''Vector''' is a quantity that has both '''magnitude''' and '''direction'''.  
  
 
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An '''identity matrix''' is a '''square matrix'''.  
 
An '''identity matrix''' is a '''square matrix'''.  
  
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It has all the diagonal elements as 1 and rest all elements as 0.  
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|| A*C={{15,23},{20,28},{9,14}}  
 
|| A*C={{15,23},{20,28},{9,14}}  
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|| Product of '''matrices A''' and C is displayed as '''M2''' in the '''Algebra view'''.  
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|| Product of '''matrices A''' and '''C''' is displayed as '''M2''' in the '''Algebra view'''.  
  
 
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Revision as of 16:09, 9 May 2018

Time Narration
Slide Number 1

Title Slide

Welcome to this tutorial on Vectors and Matrices in Geogebra.
Slide Number 2

Learning Objectives

In this tutorial, we will learn about,

How to draw a vector

Arithmetic operations on vectors

How to create a matrix

Arithmetic operations on matrices

Transpose of a matrix

Determinant of a matrix

Inverse of a matrix

Slide Number 3

System Requirement

Here I am using,

Ubuntu Linux OS version 14.04

GeoGebra version 5.0.388.0-d.

Slide Number 4

Pre requisites

www.spoken-tutorial.org.

To follow this tutorial, you should be familiar with Geogebra interface.

If not, for relevant Geogebra tutorials please visit our website.

Let us define a vector.
Slide Number 5

Vector

Vector is a quantity that has both magnitude and direction.
Cursor on GeoGebra window. I have opened a GeoGebra window.
Go to Options >> Font Size.

From Sub-menu >> 20 pt(point) radio button.

Before I start this demonstration I will change the font size to 20.


Go to Options menu, scroll down to Font Size.


From the sub-menu select 20 pt(point) radio button.

Click on Vector tool,

Click on origin >> drag to draw a vector u. (draw the vector with angle less than 90 degree)

Let us draw a vector.

Click on Line tool drop down and select Vector tool.

Click on the Origin(0,0) and drag the mouse to draw a vector u.

Click on Vector tool,

Click on Origin >> drag to draw a vector u. (draw the vector with angle less than 90 degree.)

Let us draw another vector v from the origin.
Cursor on Graphics view. Let us show the relation between vectors and a parallelogram.
Slide Number 6

Parallelogram Law of Vector Addition

Consider two vectors as two adjacent sides of a parallelogram.

Then resultant of these vectors is the diagonal of the parallelogram.

Point to input bar.

Type u+v >> press enter.

Point to vector w in Graphics view and Algebra view.

Let's add the vectors u and v.

In the input bar, type u+v and press Enter.

Here vector w, represents addition of the vectors u and v.

Cursor on Graphics view. Let's show that vector w is diagonal of the parallelogram.
Cursor on the Graphics view. To demonstrate this, let's complete the parallelogram.
Click on Vector from Point tool.

Click on point B >> vector v.

Point to the new vector.

Click on the Line drop-down and select Vector from Point tool.


Click on point B and vector v.

The new vector a same as vector v is drawn.

Click point C >> click vector u.

point to vector b.

Now click on point C and vector u .

The new vector b same as vector u, is drawn.

Click on Move tool >> drag B' . Using Move tool move the labels.
Point to the parallelogram ABB'C.

Point to the diagonal AB' .

Parallelogram ABB'C is completed.

Notice that diagonal AB' represents sum of vectors u and v.

Press CTRL+Z Press CTRL+Z to undo the process.

Retain the vector u.

Point to vector u. Now we have vector u on Graphics view.
Point to the coordinates of the vector. Cartesian coordinates of the vector are shown in the Algebra view.
Point to the values.


Move point B using Move tool.

Here values of magnitude and angle of vector u are displayed.

If we move point B, values change accordingly.

Point to the Algebra view.

Right click on the vector.

Click on Polar coordinates.

In the Algebra view, right click on vector u.

A sub-menu appears.

Select Polar coordinates.

Observe the coordinates in the polar form.

Right click on point B.

Click on Polar coordinates.

To change the values manually, right click on point B.

Select Polar coordinates.

Double click to change the values.

Type 5 as magnitude; 50 as angle, press Enter. (5; 50)

Point to the vector.

Double-click on point B to change the values.

Type 5 as magnitude; 50 as angle and press Enter.

Notice the change in magnitude and angle of vector u.

Let us multiply a vector by a scalar.
Type 2u in the input bar >> press Enter.

Type -2u >> press Enter.

Point to the vectors.

Type 2u in the input bar and press Enter.


The magnitude of new vector is equal to 2u.

Type -2u and press Enter.

The magnitude of new vector is 2u, but in opposite direction.

Point to the Zoom Out tool. To view the new vectors, use Zoom Out tool from tool bar.
Slide Number 7

Assignment

Ex: u/3.

As an assignment,

1. Subtract the vectors u and v

2. Divide a vector by a scalar.

Now we will move on to matrices.
Slide Number 8

Matrix

mxn matrix

A matrix is an ordered set of numbers.


It is listed in a rectangular form as ‘m’ rows and ‘n’ columns.

Slide Number 9


Identity Matrix

A unit matrix is I=[1].

It has m=n=1 and element is also 1.

An identity matrix is a square matrix.

It has all the diagonal elements as 1 and rest of the elements as 0.

Slide Number 10


Identity Matrix

X= [1 0, 1 0] is 2 by 2 identity matrix and

Y=[1 0 0, 0 1 0, 0 0 1] is 3 by 3 identity matrix.

Slide Number 11


Create Matrices

In GeoGebra, we can create a matrix using:

Spreadsheet view

CAS view and

Input bar.

File >> New Window. Let's open a new window.
Go to View menu >> click Spreadsheet check box. To create matrices, we will close Graphics view and open Spreadsheet view.
Type the elements of the matrix.

A= {{1, 3, 2},{2,4,0},{ 1,0,5}}

Type the elements of the matrix in the spreadsheet.
Type the elements in A1.


Type 1 3 2.

Type the elements in the cells starting from A1.


Type the first row elements as 1 3 2.

Type elements.

2 4 0 >> 1 0 5.

Similarly type the remaining elements.
Select the matrix elements.

Click on Matrix.

To create a matrix, select the matrix elements.

Click on List drop-down and select Matrix.

Point to the dialog box. Matrix dialog-box opens.
Point to Name text box.

Type the name of the matrix as A.

Click on Create button.

Point to the matrix.

In the Name text box, type the name of matrix as A.

Click on Create button.

A 3 by 3 matrix is displayed in the Algebra view.

Go to View menu click on CAS check box. Let us create the same matrix using CAS view.

To open CAS view, go to View menu, click on CAS check box.

{{1, 3, 2},{2,4,0},{ 1,0,5}} In the first box, type the elements of the matrix as shown and press Enter.

Here, inner curly brackets represent different rows.

Click on X. Close the CAS view.
Point to the Algebra view.

B={{2,4, 6},{4,2,3},{5,3,4}}

Similarly, we will create another 3 by 3 matrix B.

Type the elements of the matrix in the spreadsheet as shown.

Select the elements >> right click .

Point to sub-menu.

To create a matrix, select the elements and right click.

A sub-menu opens.

Select Create >> select Matrix. Navigate to Create and select Matrix.
Right click on the matrix in Algebra view>> select Rename. To rename the matrix, right click on the matrix in the Algebra View.

Select Rename.

Rename dialog box appears. Rename dialog-box appears.
Type the name as B >> click on OK. Type the name as B and click OK.
Addition/Subtraction of Matrices. We can add or subtract matrices only if they are of the same order.
Cursor on Algebra view. Now we will add the matrices A and B.
Point to input bar.

Type A + B in input bar >> press Enter.

In the input bar, type A + Band press Enter.
Point to the Algebra view

A+B={{3,7,8},{6,6,3},{6,3,9}}

Addition matrix M1 is displayed in the Algebra view.
Now we will see multiplication of matrices.
Slide Number 12

Matrix Multiplication

Two matrices X and Y can be multiplied if,

number of columns of X is equal to the number of rows of Y.

X is m by n matrix, Y is n by p matrix.

X into Y is a matrix of order m by p.

Point to matrix C.

C={{4,4},{3,5},{1,2}}

Let us will create a 3 by 2 matrix C using the input bar.


In the input bar, type the matrix C as shown and press Enter.

Let us multiply the matrices A and C.
Point to input bar.

In input bar, type, A*C (asterisk) >>press Enter.

In the input bar, type, A asterisk C and press Enter.
A*C={{15,23},{20,28},{9,14}} Product of matrices A and C is displayed as M2 in the Algebra view.
Slide Number 13

Assignment

As an assignment,

1. Subtract matrices

2. Multiply matrices of same order and different order.

In input bar, type, transpose.

Select Transpose[Matrix]

To show transpose of matrix A- in the input bar, type: transpose.

Select Transpose[Matrix]

Type A in place of Matrix >> press Enter.

Transpose[A]={{1,3,2},{2,4,0}{1,0,5}}

Type A in place of Matrix and press Enter.
Transpose[A]= {{1,2,1},{3,4,0},{2,0,5}} Transpose of a matrix M3 is displayed in the Algebra view.
Point to matrix A. Now, we will show determinant of matrix A.
Point to the input bar.

Type, determinant

Select Determinant[Matrix]


Type A in place of Matrix >> press Enter.

In the input bar, type determinant

Select Determinant[Matrix]

Type A in place of Matrix and press Enter.

Point to the determinant value.

Determinant[A]=-18

Value of Determinant of matrix A is displayed in the Algebra view.
Slide Number 14


Inverse of a Matrix

A square matrix P has an inverse, only if the determinant of P is not equal to zero (|P|≠0).
In the input bar, type, invert

Select Invert[Matrix]

Now, we show inverse of matrix A.

In the input bar, type, invert

Select Invert[Matrix]

Type A in place of Matrix >> press Enter. Type A in place of Matrix and press Enter.
Point to inverse of A.

Invert[A]={{-1.11, 0.83, 0.44},{0.56,-0.17,-0.22},{0.22, -0.17, 0.11}}

Drag the border of Algebra view to see the inverse matrix

Inverse of matrix A, M4 is displayed in the Algebra view.

Cursor on the Spreadsheet view. If determinant value of a matrix is zero, its inverse does not exist.

For this we will create a new matrix D.

D={{1,2,3},{4,5,6},{7,8,9}} Type the elements of the matrix as shown.
Select the elements >> right click.

Sub-menu opens.

Select the elements and right click to open a sub-menu.
Select Create >> select Matrix. Select Create and then select Matrix.
Right-click on M5 in the Algebra view.

Select Rename from the sub-menu.

Type D in the Rename text box.

Rename the matrix M5 in the Algebra view as D.
Type, determinant

Select Determinant[Matrix]

Using the input bar, let us find the determinant.

Type determinant

Select Determinant[Matrix]

Type D in place of Matrix >> press Enter. Type D in place of Matrix and press Enter.
Point to Algebra view. We see that determinant of matrix D is zero.
In the input bar, type, Invert(D) >> press Enter. Now, in the input bar, type, Invert(D)

and press Enter.

Point to L1 undefined in the Algebra view. L1 undefined is displayed in the Algebra view.

This indicates that inverse of matrix D cannot be determined.

Slide Number 15


Assignment

As an assignment,


Find the determinant and inverse of Matrices B and C.

Let's summarize.
Slide Number 16

Summary

In this tutorial, we have learnt,

How to draw a vector

Arithmetic operations on vectors

How to create a matrix

Arithmetic operations on matrices

Transpose of a matrix

Determinant of a matrix

Inverse of a matrix .

Slide Number 17

About Spoken Tutorial project

The video at the following link summarises the Spoken Tutorial project.

Please download and watch it.

Slide Number 18

Spoken Tutorial workshops

The Spoken Tutorial Project team:

conducts workshops using spoken tutorials and

gives certificates on passing online tests.

For more details, please write to us.

Slide Number 19

Forum for specific questions:

Do you have questions in THIS Spoken Tutorial?

Please visit this site

Choose the minute and second where you have the question

Explain your question briefly

Someone from our team will answer them.

Slide Number 20

Forum for specific questions:

The Spoken Tutorial forum is for specific questions on this tutorial

Please do not post unrelated and general questions on them

This will help reduce the clutter

With less clutter, we can use these discussion as instructional material.

Slide Number 21

Acknowledgement

Spoken Tutorial Project is funded by NMEICT, MHRD, Government of India.

More information on this mission is available at this link.

This is Madhuri Ganapathi from, IIT Bombay signing off.

Thank you for watching.

Contributors and Content Editors

Karwanjehimanshi95, Madhurig, Nancyvarkey

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