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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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