Applications-of-GeoGebra/C2/Vectors-and-Matrices/English-timed
From Script | Spoken-Tutorial
Time | Narration |
00:01 | Welcome to this tutorial on Vectors and Matrices in Geogebra. |
00:06 | In this tutorial, we will learn about,
How to draw a vector |
00:11 | Arithmetic operations on vectors |
00:14 | How to create a matrix |
00:16 | Arithmetic operations on matrices |
00:19 | Transpose of a matrix |
00:22 | Determinant of a matrix |
00:25 | Inverse of a matrix |
00:28 | Here I am using, Ubuntu Linux OS version 14.04
GeoGebra version 5.0.388.0 hyphen d. |
00:40 | To follow this tutorial, you should be familiar with Geogebra interface. |
00:47 | If not, for relevant Geogebra tutorials please visit our website. |
00:52 | Let us define a vector. |
00:55 | Vector is a quantity that has both magnitude and direction. |
01:00 | I have opened a GeoGebra window. |
01:03 | Before I start this demonstration I will change the font size to 20. |
01:08 | Go to Options menu, scroll down to Font Size. |
01:12 | From the sub-menu select 20 point radio button. |
01:16 | Let us draw a vector. |
01:19 | Click on Line tool drop down and select Vector tool. |
01:25 | Click on the Origin(0,0) and drag the mouse to draw a vector u. |
01:30 | Let us draw another vector v from the origin. |
01:34 | Let us show the relation between vectors and a parallelogram. |
01:39 | Consider two vectors as two adjacent sides of a parallelogram. |
01:43 | Then the resultant of these vectors is the diagonal of the parallelogram. |
01:48 | Let's add the vectors u and v. |
01:51 | In the input bar, type u+v and press Enter. |
01:57 | Here vector w, represents addition of the vectors u and v. |
02:02 | Let's show that vector w is diagonal of the parallelogram. |
02:07 | To demonstrate this, let's complete the parallelogram. |
02:11 | Click on the Line drop-down and select Vector from Point tool. |
02:17 | Click on point B and vector v. |
02:21 | The new vector a same as vector v is drawn. |
02:25 | Click on point C and vector u . |
02:29 | The new vector b same as vector u, is drawn. |
02:33 | Using Move tool, move the labels. |
02:37 | Parallelogram A B Bdash C is completed. |
02:42 | Notice that diagonal A Bdash represents sum of vectors u and v. |
02:48 | Press CTRL+Z to undo the process. |
02:53 | Retain the vector u. |
02:55 | Now we have vector u on Graphics view. |
02:59 | Cartesian coordinates of the vector are shown in the Algebra view. |
03:04 | Here values of magnitude and angle of vector u are displayed. |
03:10 | If we move point B, values change accordingly. |
03:15 | In the Algebra view, right click on vector u. |
03:19 | A sub-menu appears.
Select Polar coordinates. |
03:24 | Observe the coordinates in the polar form. |
03:27 | To change the values manually, right click on point B. |
03:31 | Select Polar coordinates. |
03:34 | Double-click on point B to change the values. |
03:38 | Type 5 as magnitude; 50 as angle and press Enter. |
03:45 | Notice the change in magnitude and angle of vector u. |
03:49 | Let us multiply a vector by a scalar. |
03:53 | Type 2u in the input bar and press Enter. |
03:57 | The magnitude of new vector is equal to 2u. |
04:01 | Type minus 2u and press Enter. |
04:05 | The magnitude of new vector is 2u, but in opposite direction. |
04:10 | To view the new vectors, use Zoom Out tool from tool bar. |
04:17 | As an assignment,
Subtract the vectors u and v |
04:22 | Divide a vector by a scalar. |
04:25 | Now we will move on to matrices. |
04:28 | A matrix is an ordered set of numbers. |
04:31 | It is listed in a rectangular form as ‘m’ rows and ‘n’ columns. |
04:36 | A unit matrix is I equal to 1 |
04:40 | It has m equal to n equal to 1 and element is also 1. |
04:47 | An identity matrix is a square matrix. |
04:51 | It has all the diagonal elements as 1 and rest of the elements as 0. |
04:56 | X is a 2 by 2 identity matrix and |
05:00 | Y is a 3 by 3 identity matrix. |
05:04 | In GeoGebra, we can create a matrix using:
Spreadsheet view , CAS view and Input bar. |
05:13 | Let's open a new window. |
05:18 | To create matrices, we will close Graphics view and open Spreadsheet view. |
05:26 | Type the elements of the matrix in the spreadsheet. |
05:30 | Type the elements in the cells starting from A1. |
05:34 | Type the first row elements as 1 3 2. |
05:42 | Similarly type the remaining elements. |
05:47 | To create a matrix, select the matrix elements. |
05:51 | Click on List drop-down and select Matrix. |
05:56 | Matrix dialog-box opens. |
05:59 | In the Name text box, type the name of matrix as A. |
06:04 | Click on Create button. |
06:07 | A 3 by 3 matrix is displayed in the Algebra view. |
06:11 | Let us create the same matrix using CAS view. |
06:15 | To open CAS view, go to View menu, click on CAS check box. |
06:23 | In the first box, type the elements of the matrix as shown and press Enter. |
06:30 | Here, inner curly brackets represent different rows. |
06:35 | Close the CAS view. |
06:37 | Similarly, we will create another 3 by 3 matrix B. |
06:42 | Type the elements of the matrix in the spreadsheet as shown. |
06:46 | To create a matrix, select the elements and right click. |
06:51 | A sub-menu opens. |
06:53 | Navigate to Create and select Matrix. |
06:58 | To rename the matrix, right click on the matrix in the Algebra View. |
07:03 | Select Rename. |
07:05 | Rename dialog-box appears. |
07:08 | Type the name as B and click OK. |
07:14 | We can add or subtract matrices only if they are of the same order. |
07:19 | Now we will add the matrices A and B. |
07:22 | In the input bar, type A + Band press Enter. |
07:28 | Addition matrix M1 is displayed in the Algebra view. |
07:32 | Now we will see multiplication of matrices. |
07:36 | Two matrices X and Y can be multiplied if, |
07:40 | number of columns of X is equal to the number of rows of Y. |
07:45 | X is m by n matrix, Y is n by p matrix. |
07:50 | X into Y is a matrix of order m by p. |
07:54 | Let us will create a 3 by 2 matrix C using the input bar. |
07:59 | In the input bar, type the matrix C as shown and press Enter. |
08:06 | Let us multiply the matrices A and C. |
08:10 | In the input bar, type, A asterisk C and press Enter. |
08:16 | Product of matrices A and C is displayed as M2 in the Algebra view. |
08:22 | As an assignment,
Subtract matrices , Multiply matrices of same order and different order. |
08:30 | To show transpose of matrix A- in the input bar, type: transpose.
Select Transpose Matrix |
08:38 | Type A in place of Matrix and press Enter. |
08:42 | Transpose of a matrix M3 is displayed in the Algebra view. |
08:47 | Now, we will show determinant of matrix A. |
08:51 | In the input bar, type determinant |
08:54 | Select Determinant Matrix |
08:57 | Type A in place of Matrix and press Enter. |
09:01 | Value of Determinant of matrix A is displayed in the Algebra view. |
09:06 | A square matrix P has an inverse, only if the determinant of P is not equal to zero |
09:13 | Now, we show inverse of matrix . |
09:16 | In the input bar, type, invert |
09:19 | Select Invert Matrix |
09:22 | Type A in place of Matrix and press Enter. |
09:26 | Drag the border of Algebra view to see the inverse matrix |
09:31 | Inverse of matrix A, M4 is displayed in the Algebra view. |
09:36 | If determinant value of a matrix is zero, its inverse does not exist. |
09:41 | For this we will create a new matrix D. |
09:45 | Type the elements of the matrix as shown. |
09:49 | Select the elements and right click to open a sub-menu. |
09:53 | Select Create and then select Matrix. |
09:58 | Rename the matrix M5 in the Algebra view as D. |
10:03 | Using the input bar, let us find the determinant. |
10:07 | Type determinant |
10:09 | Select Determinant Matrix |
10:12 | Type D in place of Matrix and press Enter. |
10:16 | We see that determinant of matrix D is zero. |
10:20 | Now, in the input bar, type, Invert(D)
and press Enter. |
10:26 | L1 undefined is displayed in the Algebra view. |
10:30 | This indicates that inverse of matrix D cannot be determined. |
10:36 | As an assignment,
Find the determinant and inverse of Matrices B and C. |
10:43 | Let's summarize. |
10:45 | In this tutorial, we have learnt,
How to draw a vector |
10:49 | Arithmetic operations on vectors |
10:52 | How to create a matrix |
10:54 | Arithmetic operations on matrices |
10:58 | Transpose of a matrix |
11:01 | Determinant of a matrix |
11:04 | Inverse of a matrix . |
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12:08 | This is Madhuri Ganapathi from, IIT Bombay signing off.
Thank you for watching. |