Applications-of-GeoGebra/C2/Vectors-and-Matrices/English-timed

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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