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

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(Created page with "{| border=1 ||'''Time''' ||'''Narration''' |- ||'''Slide Number 1''' '''Title Slide''' || Welcome to this tutorial on '''Vectors and Matrices''' in '''Geogebra'''....")
 
 
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{| border=1
 
{| border=1
−
||'''Time'''   
+
||'''Visual Cue'''   
 
||'''Narration'''   
 
||'''Narration'''   
 
|-
 
|-
Line 14: Line 14:
 
|| In this tutorial, we will learn about,  
 
|| In this tutorial, we will learn about,  
  
−
How to draw a vector  
+
How to draw a '''vector'''
  
−
Arithmetic operations on vectors  
+
Arithmetic operations on '''vectors'''
  
−
How to create a matrix  
+
How to create a '''matrix'''
  
−
Arithmetic operations on matrices
+
Arithmetic operations on '''matrices'''
  
−
Transpose of a matrix  
+
'''Transpose''' of a '''matrix'''
  
−
Determinant of a matrix  
+
'''Determinant''' of a '''matrix'''
  
−
Inverse of a matrix  
+
'''Inverse''' of a '''matrix'''
  
 
|-  
 
|-  
Line 44: Line 44:
  
 
'''www.spoken-tutorial.org'''.  
 
'''www.spoken-tutorial.org'''.  
−
|| To follow this tutorial, you should be familiar with,
+
|| To follow this tutorial, you should be familiar with '''Geogebra''' interface.  
−
 
+
−
'''Geogebra''' interface.  
+
  
 
If not, for relevant '''Geogebra''' tutorials please visit our website.  
 
If not, for relevant '''Geogebra''' tutorials please visit our website.  
Line 52: Line 50:
 
|-  
 
|-  
 
||  
 
||  
−
|| Let’s define a '''vector'''.  
+
|| Let us define a '''vector'''.  
  
 
|-  
 
|-  
Line 58: Line 56:
  
 
'''Vector'''  
 
'''Vector'''  
−
|| A vector is a quantity that has both magnitude and direction.  
+
|| '''Vector''' is a quantity that has both '''magnitude''' and '''direction'''.  
  
 
|-  
 
|-  
Line 80: Line 78:
  
 
Click on origin >> drag to draw a vector '''u'''. (draw the vector with angle less than 90 degree)  
 
Click on origin >> drag to draw a vector '''u'''. (draw the vector with angle less than 90 degree)  
−
|| Let's draw a vector.  
+
|| Let us draw a '''vector'''.  
  
 
Click on '''Line tool''' drop down and select '''Vector''' tool.  
 
Click on '''Line tool''' drop down and select '''Vector''' tool.  
Line 90: Line 88:
  
 
Click on Origin >> drag to draw a vector '''u'''. (draw the vector with angle less than 90 degree.)  
 
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.  
+
| | Let us draw another '''vector v''' from the origin.  
 
|-  
 
|-  
 
| | Cursor on '''Graphics view'''.  
 
| | Cursor on '''Graphics view'''.  
−
| | Let us show the relation between vectors and a parallelogram.  
+
| | Let us show the relation between '''vectors''' and a '''parallelogram'''.  
  
 
|-  
 
|-  
Line 100: Line 98:
 
'''Parallelogram Law of Vector Addition'''  
 
'''Parallelogram Law of Vector Addition'''  
 
   
 
   
−
|| Consider two vectors as two adjacent sides of a '''parallelogram. '''  
+
|| Consider two '''vectors''' as two adjacent sides of a '''parallelogram. '''  
  
−
Then resultant of these vectors is the diagonal of the '''parallelogram'''.  
+
Then resultant of these '''vectors''' is the diagonal of the '''parallelogram'''.  
  
 
|-  
 
|-  
Line 112: Line 110:
 
|| Let's add the vectors '''u''' and '''v'''.  
 
|| Let's add the vectors '''u''' and '''v'''.  
  
−
In the input bar, type '''u+v''' and press Enter.  
+
In the input bar, type '''u+v''' and press '''Enter'''.  
  
−
Here vector '''w''', represents addition of the vectors '''u''' and '''v'''.  
+
Here '''vector w''', represents addition of the '''vectors u''' and '''v'''.  
  
 
|-  
 
|-  
 
|| Cursor on '''Graphics view.'''  
 
|| Cursor on '''Graphics view.'''  
−
|| Let's show that vector '''w''' is diagonal of the parallelogram.  
+
|| Let's show that '''vector w''' is '''diagonal''' of the '''parallelogram'''.  
  
 
|-  
 
|-  
 
|| Cursor on the '''Graphics view'''.  
 
|| Cursor on the '''Graphics view'''.  
−
|| To demonstrate this, let's complete the parallelogram.  
+
|| To demonstrate this, let's complete the '''parallelogram'''.  
  
 
|-  
 
|-  
Line 133: Line 131:
  
  
−
Click on point '''B''' and vector '''v'''.  
+
Click on point '''B''' and '''vector v'''.  
  
−
The new vector '''a''' same as vector '''v ''' is drawn.  
+
The new '''vector a''' same as '''vector v ''' is drawn.  
  
 
|-  
 
|-  
Line 141: Line 139:
  
 
point to vector '''b'''.  
 
point to vector '''b'''.  
−
|| Now click point '''C''' and vector '''u''' .
+
|| Now click on point '''C''' and '''vector u''' .
  
−
The new vector '''b''' same as vector '''u''' is drawn.  
+
The new '''vector b''' same as vector '''u,''' is drawn.  
 
|-  
 
|-  
 
|| Click on '''Move''' tool >> drag '''B' '''.  
 
|| Click on '''Move''' tool >> drag '''B' '''.  
Line 152: Line 150:
  
 
Point to the diagonal '''AB' '''.  
 
Point to the diagonal '''AB' '''.  
−
|| Parallelogram '''ABB'C ''' is completed.  
+
|| '''Parallelogram ABB'C ''' is completed.  
  
−
Notice that diagonal '''AB' ''' represents sum of vectors '''u''' and '''v'''.  
+
Notice that '''diagonal AB' ''' represents sum of '''vectors u''' and '''v'''.  
  
 
|-  
 
|-  
Line 160: Line 158:
 
|| Press '''CTRL+Z''' to undo the process.  
 
|| Press '''CTRL+Z''' to undo the process.  
  
−
Retain the vector '''u'''.  
+
Retain the '''vector u'''.  
  
 
|-  
 
|-  
 
|| Point to vector '''u'''.  
 
|| Point to vector '''u'''.  
−
|| Now we have vectors '''u''' on '''Graphics view'''.  
+
|| Now we have '''vector u''' on '''Graphics view'''.  
  
 
|-  
 
|-  
 
|| Point to the coordinates of the vector.  
 
|| Point to the coordinates of the vector.  
−
|| '''Cartesian '''coordinates''' of the vector are shown in the '''Algebra view'''.  
+
|| '''Cartesian coordinates''' of the '''vector''' are shown in the '''Algebra view'''.  
  
 
|-  
 
|-  
Line 175: Line 173:
  
 
Move point '''B''' using '''Move''' tool.  
 
Move point '''B''' using '''Move''' tool.  
−
|| Here values of magnitude and angle of vector '''u''' are displayed.  
+
|| Here values of '''magnitude''' and angle of '''vector u''' are displayed.  
  
 
If we move point '''B''', values change accordingly.  
 
If we move point '''B''', values change accordingly.  
Line 185: Line 183:
  
 
Click on '''Polar coordinates'''.  
 
Click on '''Polar coordinates'''.  
−
|| In the '''Algebra view,''' right click on vector '''u'''.  
+
|| In the '''Algebra view,''' right click on '''vector u'''.  
  
 
A sub-menu appears.  
 
A sub-menu appears.  
Line 191: Line 189:
 
Select '''Polar coordinates'''.  
 
Select '''Polar coordinates'''.  
  
−
Observe the coordinates in the '''polar''' form.  
+
Observe the '''coordinates''' in the '''polar''' form.  
  
 
|-  
 
|-  
Line 202: Line 200:
  
 
|-  
 
|-  
−
|| double click to change the values.  
+
|| Double click to change the values.  
 
   
 
   
 
Type '''5''' as magnitude; '''50''' as angle, press '''Enter'''. (5; 50)  
 
Type '''5''' as magnitude; '''50''' as angle, press '''Enter'''. (5; 50)  
Line 209: Line 207:
 
|| Double-click on point '''B''' to change the values.  
 
|| Double-click on point '''B''' to change the values.  
  
−
Type '''5''' as magnitude; '''50''' as angle and press '''Enter'''.  
+
Type '''5''' as '''magnitude'''; '''50''' as angle and press '''Enter'''.  
  
−
Notice the change in magnitude and angle of vector '''u'''.  
+
Notice the change in '''magnitude''' and angle of '''vector u'''.  
  
 
|-  
 
|-  
 
||  
 
||  
−
|| Let us multiply a vector by a scalar.  
+
|| Let us multiply a '''vector''' by a '''scalar'''.  
  
 
|-  
 
|-  
Line 226: Line 224:
  
  
−
The magnitude of new vector is equal to 2u.  
+
The '''magnitude''' of new '''vector''' is equal to 2u.  
  
 
Type '''-2u''' and press '''Enter'''.  
 
Type '''-2u''' and press '''Enter'''.  
  
−
The magnitude of new vector is '''2u''', but in opposite direction.  
+
The '''magnitude''' of new '''vector''' is '''2u''', but in opposite direction.  
  
 
|-  
 
|-  
 
|| Point to the '''Zoom Out''' tool.  
 
|| Point to the '''Zoom Out''' tool.  
−
|| To view the new vectors, use '''Zoom Out''' tool from tool bar.  
+
|| To view the new '''vectors''', use '''Zoom Out''' tool from tool bar.  
  
 
|-  
 
|-  
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|| As an assignment,  
 
|| As an assignment,  
  
−
1. Subtract the vectors u and v  
+
1. Subtract the '''vectors u''' and '''v'''
  
−
2. Divide a vector by a scalar.  
+
2. Divide a '''vector''' by a '''scalar'''.  
 
|-  
 
|-  
 
||  
 
||  
−
|| Now we will move on to matrices.  
+
|| Now we will move on to '''matrices'''.  
  
 
|-  
 
|-  
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|| A unit matrix is I=[1].  
 
|| A unit matrix is I=[1].  
  
−
It has m=n=1 and element is also 1.  
+
It has m=n=1 and '''element''' is also 1.  
  
−
An '''identity matrix''' is a square matrix.  
+
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.  
  
 
|-  
 
|-  
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'''Identity Matrix'''  
 
'''Identity Matrix'''  
−
|| X= [1 0, 1 0] is 2x2 identity matrix and  
+
|| X= [1 0, 1 0] is 2 by 2 '''identity matrix''' and  
  
−
Y=[1 0 0, 0 1 0, 0 0 1] is 3x3 identity matrix.  
+
Y=[1 0 0, 0 1 0, 0 0 1] is 3 by 3 '''identity matrix'''.  
  
 
|-  
 
|-  
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'''Create Matrices'''  
 
'''Create Matrices'''  
−
|| In GeoGebra, we can create a matrix using:  
+
|| In GeoGebra, we can create a '''matrix''' using:  
  
 
'''Spreadsheet view '''  
 
'''Spreadsheet view '''  
Line 303: Line 301:
 
|-  
 
|-  
 
|| Go to '''View''' menu >> click '''Spreadsheet''' check box.  
 
|| Go to '''View''' menu >> click '''Spreadsheet''' check box.  
−
|| To create matrices, we will close '''Graphics''' view and open '''Spreadsheet''' view.  
+
|| To create '''matrices''', we will close '''Graphics''' view and open '''Spreadsheet''' view.  
  
 
|-  
 
|-  
−
|| Type the elements of the matrix.  
+
|| Type the '''elements''' of the '''matrix'''.  
  
 
A= {{1, 3, 2},{2,4,0},{ 1,0,5}}  
 
A= {{1, 3, 2},{2,4,0},{ 1,0,5}}  
  
−
|| Type the elements of the matrix in the '''spreadsheet'''.  
+
|| Type the '''elements''' of the '''matrix''' in the '''spreadsheet'''.  
  
 
|-  
 
|-  
−
|| Type the elements in A1.  
+
|| Type the '''elements''' in '''A1'''.  
  
  
Line 320: Line 318:
  
  
−
Type the first row elements as 1 3 2.  
+
Type the first row '''elements''' as 1 3 2.  
  
 
|-  
 
|-  
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2 4 0 >> 1 0 5.  
 
2 4 0 >> 1 0 5.  
−
|| Similarly type the remaining elements.  
+
|| Similarly type the remaining '''elements'''.  
  
 
|-  
 
|-  
Line 332: Line 330:
  
 
Click on '''Matrix'''.  
 
Click on '''Matrix'''.  
−
|| To create a matrix, select the matrix elements.  
+
|| To create a '''matrix''', select the '''matrix elements.'''
  
 
Click on''' List''' drop-down and select '''Matrix'''.  
 
Click on''' List''' drop-down and select '''Matrix'''.  
Line 348: Line 346:
  
 
Point to the matrix.  
 
Point to the matrix.  
−
|| In the '''Name''' text box, type the name of matrix as '''A'''.  
+
|| In the '''Name''' text box, type the name of '''matrix''' as '''A'''.  
  
 
Click on '''Create''' button.  
 
Click on '''Create''' button.  
  
−
A 3x3 matrix is displayed in the '''Algebra view.'''  
+
A 3 by 3 '''matrix''' is displayed in the '''Algebra view.'''  
  
 
|-  
 
|-  
 
|| Go to '''View''' menu click on '''CAS''' check box.  
 
|| Go to '''View''' menu click on '''CAS''' check box.  
−
|| Let us create the same matrix using '''CAS view'''.  
+
|| Let us create the same '''matrix''' using '''CAS view'''.  
  
 
To open '''CAS view''', go to '''View''' menu, click on '''CAS''' check box.  
 
To open '''CAS view''', go to '''View''' menu, click on '''CAS''' check box.  
Line 362: Line 360:
 
|-  
 
|-  
 
|| {{1, 3, 2},{2,4,0},{ 1,0,5}}  
 
|| {{1, 3, 2},{2,4,0},{ 1,0,5}}  
−
|| In the first box, type the elements of the matrix as shown and press '''Enter'''.  
+
|| In the first box, type the '''elements''' of the '''matrix''' as shown and press '''Enter'''.  
  
 
Here, inner curly brackets represent different rows.  
 
Here, inner curly brackets represent different rows.  
Line 374: Line 372:
  
 
B={{2,4, 6},{4,2,3},{5,3,4}}  
 
B={{2,4, 6},{4,2,3},{5,3,4}}  
−
|| Similarly, we will create another 3x3 matrix '''B'''.  
+
|| Similarly, we will create another 3 by 3 '''matrix B'''.  
  
−
Type the elements of the matrix in the '''spreadsheet''' as shown.  
+
Type the '''elements''' of the '''matrix''' in the '''spreadsheet''' as shown.  
  
 
|-  
 
|-  
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Point to sub-menu.  
 
Point to sub-menu.  
−
|| To create a matrix, select the elements and right click.  
+
|| To create a '''matrix''', select the '''elements''' and right click.  
  
 
A sub-menu opens.  
 
A sub-menu opens.  
Line 393: Line 391:
 
||Right click on the matrix in '''Algebra view'''>> select '''Rename'''.  
 
||Right click on the matrix in '''Algebra view'''>> select '''Rename'''.  
  
−
|| To rename the matrix, right click on the matrix in the '''Algebra View'''.  
+
|| To rename the '''matrix''', right click on the '''matrix''' in the '''Algebra View'''.  
  
 
Select '''Rename'''.  
 
Select '''Rename'''.  
Line 407: Line 405:
 
|-  
 
|-  
 
|| Addition/Subtraction of Matrices.  
 
|| Addition/Subtraction of Matrices.  
−
|| We can add or subtract matrices only if they are of the same order.  
+
|| We can add or subtract '''matrices''' only if they are of the same order.  
  
 
|-  
 
|-  
 
|| Cursor on Algebra view.  
 
|| Cursor on Algebra view.  
−
|| Now we will add the matrices '''A''' and '''B'''.  
+
|| Now we will add the '''matrices A''' and '''B'''.  
  
 
|-  
 
|-  
Line 417: Line 415:
  
 
Type '''A + B''' in input bar >> press '''Enter'''.  
 
Type '''A + B''' in input bar >> press '''Enter'''.  
−
|| In the input bar, type '''A + B'''and press '''Enter'''.  
+
|| In the '''input bar''', type '''A + B'''and press '''Enter'''.  
  
 
|-  
 
|-  
Line 423: Line 421:
  
 
A+B={{3,7,8},{6,6,3},{6,3,9}}  
 
A+B={{3,7,8},{6,6,3},{6,3,9}}  
−
|| Addition matrix '''M1''' is displayed in the '''Algebra view'''.  
+
|| Addition '''matrix M1''' is displayed in the '''Algebra view'''.  
  
 
|-  
 
|-  
 
||  
 
||  
−
|| Now we will see multiplication of matrices.  
+
|| Now we will see multiplication of '''matrices'''.  
  
 
|-  
 
|-  
Line 434: Line 432:
 
'''Matrix Multiplication'''  
 
'''Matrix Multiplication'''  
  
−
|| Two matrices '''X''' and '''Y '''can be multiplied if,  
+
|| Two '''matrices X''' and '''Y '''can be multiplied if,  
  
 
number of columns of '''X''' is equal to the number of rows of '''Y'''.  
 
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''' is '''m by n matrix, Y''' is '''n by p matrix'''.  
  
−
'''X*Y '''is matrix '''Z''' of order '''m × p'''.  
+
'''X into Y '''is a '''matrix ''' of order '''m by p'''.  
  
 
|-  
 
|-  
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C={{4,4},{3,5},{1,2}}  
 
C={{4,4},{3,5},{1,2}}  
−
|| Let us will create a 3x2 matrix '''C''' using the input bar.  
+
|| 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'''.  
+
In the '''input bar''', type the '''matrix C''' as shown and press '''Enter'''.  
  
 
|-  
 
|-  
 
||  
 
||  
−
|| Let us multiply the matrices A and C.  
+
|| Let us multiply the '''matrices A''' and '''C'''.  
  
 
|-  
 
|-  
 
|| Point to input bar.  
 
|| Point to input bar.  
  
−
In input bar, type, '''A*C '''(asterisk) >>press '''Enter'''.  
+
In '''input bar''', type, '''A*C '''(asterisk) >>press '''Enter'''.  
−
|| In the input bar, type, '''A*C '''(asterisk) and press '''Enter'''.  
+
|| In the '''input bar''', type, '''A asterisk C ''' and press '''Enter'''.  
  
 
|-  
 
|-  
 
|| A*C={{15,23},{20,28},{9,14}}  
 
|| 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 472: Line 470:
 
|| As an assignment,  
 
|| As an assignment,  
  
−
1. Subtract matrices  
+
1. Subtract '''matrices'''
  
−
2. Multiply matrices of same order and different order.  
+
2. Multiply '''matrices''' of same order and different order.  
  
 
|-  
 
|-  
Line 480: Line 478:
  
 
Select '''Transpose[Matrix]'''  
 
Select '''Transpose[Matrix]'''  
−
|| To show '''transpose''' of matrix '''A'''-  in the input bar, type: '''transpose'''.  
+
|| To show '''transpose''' of '''matrix A'''-  in the '''input bar''', type: '''transpose'''.  
  
 
Select '''Transpose[Matrix]'''  
 
Select '''Transpose[Matrix]'''  
Line 491: Line 489:
 
|-  
 
|-  
 
|| Transpose[A]= {{1,2,1},{3,4,0},{2,0,5}}  
 
|| Transpose[A]= {{1,2,1},{3,4,0},{2,0,5}}  
−
|| Transpose of a matrix '''M3''' is displayed in the '''Algebra view'''.  
+
|| Transpose of a '''matrix M3''' is displayed in the '''Algebra view'''.  
  
 
|-  
 
|-  
 
|| Point to matrix '''A.'''  
 
|| Point to matrix '''A.'''  
  
−
|| Now, we will show '''determinant''' of matrix '''A'''.  
+
|| Now, we will show '''determinant''' of '''matrix A'''.  
  
 
|-  
 
|-  
Line 517: Line 515:
  
 
'''Determinant[A]=-18'''  
 
'''Determinant[A]=-18'''  
−
|| Value of '''Determinant''' of matrix '''A''' is displayed in the '''Algebra view'''.  
+
|| Value of '''Determinant''' of '''matrix A''' is displayed in the '''Algebra view'''.  
  
 
|-  
 
|-  
Line 524: Line 522:
  
 
'''Inverse of a Matrix'''  
 
'''Inverse of a Matrix'''  
−
|| A square matrix '''P ''' has an '''inverse,''' only if the '''determinant''' of '''P''' is not equal to zero '''(|P|≠0)'''.  
+
|| A '''square matrix P ''' has an '''inverse,''' only if the '''determinant''' of '''P''' is not equal to zero '''(|P|≠0)'''.  
  
 
|-  
 
|-  
Line 530: Line 528:
  
 
Select '''Invert[Matrix]'''  
 
Select '''Invert[Matrix]'''  
−
|| Now, we show '''inverse''' of matrix '''A'''.  
+
|| Now, we show '''inverse''' of '''matrix A'''.  
  
−
In the input bar, type, '''invert'''  
+
In the '''input bar''', type, '''invert'''  
  
 
Select '''Invert[Matrix]'''  
 
Select '''Invert[Matrix]'''  
Line 546: Line 544:
 
|| Drag the border of  '''Algebra view''' to see the inverse matrix  
 
|| Drag the border of  '''Algebra view''' to see the inverse matrix  
  
−
Inverse of matrix '''A''', '''M4''' is displayed in the '''Algebra view.'''  
+
Inverse of '''matrix A''', '''M4''' is displayed in the '''Algebra view.'''  
  
 
|-  
 
|-  
 
|| Cursor on the '''Spreadsheet view'''.  
 
|| Cursor on the '''Spreadsheet view'''.  
−
|| If '''determinant''' value of a matrix is zero, its '''inverse''' does not exist.  
+
|| If '''determinant''' value of a '''matrix''' is zero, its '''inverse''' does not exist.  
  
−
For this we will create a new matrix '''D'''.  
+
For this we will create a new '''matrix D'''.  
  
 
|-  
 
|-  
 
|| D={{1,2,3},{4,5,6},{7,8,9}}  
 
|| D={{1,2,3},{4,5,6},{7,8,9}}  
−
|| Type the elements of the matrix as shown.  
+
|| Type the '''elements''' of the '''matrix''' as shown.  
  
 
|-  
 
|-  
Line 562: Line 560:
  
 
Sub-menu opens.  
 
Sub-menu opens.  
−
|| Select the elements and right click to open a sub-menu.  
+
|| Select the '''elements''' and right click to open a sub-menu.  
  
 
|-  
 
|-  
Line 574: Line 572:
  
 
Type '''D''' in the '''Rename''' text box.  
 
Type '''D''' in the '''Rename''' text box.  
−
|| Rename the matrix''' M5''' in the Algebra view as '''D'''.  
+
|| Rename the '''matrix M5''' in the '''Algebra view''' as '''D'''.  
  
 
|-  
 
|-  
Line 580: Line 578:
  
 
Select '''Determinant[Matrix]'''  
 
Select '''Determinant[Matrix]'''  
−
|| Using the input bar, let us find the determinant.  
+
|| Using the '''input bar''', let us find the '''determinant'''.  
  
 
Type '''determinant'''  
 
Type '''determinant'''  
Line 592: Line 590:
 
|-  
 
|-  
 
|| Point to '''Algebra view'''.  
 
|| Point to '''Algebra view'''.  
−
|| We see that '''determinant''' of matrix '''D''' is zero.  
+
|| We see that '''determinant''' of '''matrix D''' is zero.  
  
 
|-  
 
|-  
Line 604: Line 602:
 
|| '''L1 undefined''' is displayed in the '''Algebra view'''.  
 
|| '''L1 undefined''' is displayed in the '''Algebra view'''.  
  
−
This indicates that inverse of matrix '''D''' cannot be determined.  
+
This indicates that inverse of '''matrix D''' cannot be determined.  
 
|-  
 
|-  
 
|| '''Slide Number 15'''  
 
|| '''Slide Number 15'''  
Line 613: Line 611:
  
 
   
 
   
−
Find the determinant and inverse of Matrices '''B ''' and '''C'''.  
+
Find the '''determinant''' and '''inverse''' of '''Matrices B ''' and '''C'''.  
  
 
|-  
 
|-  
Line 625: Line 623:
 
|| In this tutorial, we have learnt,  
 
|| In this tutorial, we have learnt,  
  
−
How to draw a vector  
+
How to draw a '''vector'''
  
−
Arithmetic operations on vectors  
+
Arithmetic operations on '''vectors'''
  
−
How to create a matrix  
+
How to create a '''matrix'''
  
−
Arithmetic operations on matrices  
+
Arithmetic operations on '''matrices'''
  
−
Transpose of a matrix  
+
'''Transpose''' of a '''matrix'''
  
−
Determinant of a matrix  
+
'''Determinant''' of a '''matrix'''
  
−
Inverse of a matrix .
+
'''Inverse''' of a '''matrix''' .
  
 
|-  
 
|-  

Latest revision as of 11:21, 3 December 2019

Visual Cue 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

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Slide Number 18

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Slide Number 19

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Slide Number 20

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Slide Number 21

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Contributors and Content Editors

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