Karwanjehimanshi95 at 05:51, 3 December 2019

2019-12-03T05:51:29Z

Madhurig at 10:39, 9 May 2018

2018-05-09T10:39:14Z

Nancyvarkey at 07:02, 9 May 2018

2018-05-09T07:02:30Z

Madhurig: Created page with "{| border=1 ||'''Time''' ||'''Narration''' |- ||'''Slide Number 1''' '''Title Slide''' || Welcome to this tutorial on '''Vectors and Matrices''' in '''Geogebra'''...."

2018-05-08T11:58:15Z

Created page with "{| border=1 ||'''Time''' ||'''Narration''' |- ||'''Slide Number 1''' '''Title Slide''' || Welcome to this tutorial on '''Vectors and Matrices''' in '''Geogebra'''...."

New page

{| border=1
||'''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’s define a '''vector'''.

|-
|| '''Slide Number 5'''

'''Vector'''
|| A 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's 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 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 vectors '''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 all elements as 0.

|-
|| '''Slide Number 10'''

'''Identity Matrix'''
|| X= [1 0, 1 0] is 2x2 identity matrix and

Y=[1 0 0, 0 1 0, 0 0 1] is 3x3 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 3x3 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 3x3 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 + B'''and 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 × n''' matrix, '''Y''' is '''n × p '''matrix.

'''X*Y '''is matrix '''Z''' of order '''m × p'''.

|-
|| Point to matrix '''C'''.

C={{4,4},{3,5},{1,2}}
|| Let us will create a 3x2 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*C '''(asterisk) 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.
|-
|}

@@ Line 56: / Line 56: @@
 '''Vector'''
-|| A '''vector''' is a quantity that has both '''magnitude''' and '''direction'''.
+||  '''Vector''' is a quantity that has both '''magnitude''' and '''direction'''.
 |-
@@ Line 271: / Line 271: @@
 An '''identity matrix''' is a '''square matrix'''.
-It has all the diagonal elements as 1 and rest all elements as 0.
+It has all the diagonal elements as 1 and rest of the elements as 0.
 |-
@@ Line 461: / Line 461: @@
 |-
 || A*C={{15,23},{20,28},{9,14}}
-|| Product of '''matrices A''' and C is displayed as '''M2''' in the '''Algebra view'''.
+|| Product of '''matrices A''' and '''C''' is displayed as '''M2''' in the '''Algebra view'''.
 |-

@@ Line 1: / Line 1: @@
 {| border=1
-||'''Time'''
+||'''Visual Cue'''
 ||'''Narration'''
 |-

Applications-of-GeoGebra/C2/Vectors-and-Matrices/English - Revision history

Karwanjehimanshi95 at 05:51, 3 December 2019

Madhurig at 10:39, 9 May 2018

Nancyvarkey at 07:02, 9 May 2018

Madhurig: Created page with "{| border=1 ||'''Time''' ||'''Narration''' |- ||'''Slide Number 1''' '''Title Slide''' || Welcome to this tutorial on '''Vectors and Matrices''' in '''Geogebra'''...."