Geogebra 3d Vector

[Trinidad Baltzell]

Thevalue of vectors are shown in matrix form in the algebra view. A vector has a length and a direction but it has no position. Note that you can move a vector in the graphics view without changing its value.

If you right-click on a point in the algebra view you can choose to show the polar coordinates instead of the Cartesiancoordinates. GeoGebra writes polar coordinates by using a semi-colon instead if a comma.

Drag the point \(A\) and find the relations between coordinates for the other points. Create four new points by writing in the input bar. The new points should appear above the hollow points (inside each ring).

In order for the worksheet to be as user-friendly as possible, you can see to it that the mouse cursor only has a different appearance when hovered above the draggable point. Right-click on a point that isn't draggable and show its Properties. Under the tab Advanced you can uncheck "Selection Allowed".

Enter values for the number lines using a list of text objects. The command Text( , ) creates a text of an object placed at a point. By using a fifth argument to the Sequence-command, you can choose a step. Write:

Add points displaying the numbers involved on the two number lines, or slightly above or below a number line. Make vectors between points. Add dynamic text. Add other objects that help visualize the addition.

A toroidal spiral is defined by a vector function r(t)=((a+cos bt)cos t,(a+cos bt)sin t,sin bt) for some a,b.

In order to change the limits on t (currently 0 to about 4pi), use the sliders t_min and t_max near the center-top of the

screen. This also allows you to see the curve being "traced out" over time. If you want to rotate the figure, use the

sliders at the top left (theta and phi).

The equation may be adjusted by changing the radius and rate parameters on the top right. What impact do these

have on the curve? Why do the curves for integer values look so different from those with half-integer values?

[For geogebra users: The view can be rotated by adjusting theta and phi... the perspective can be similarly adjusted by changing A_x and chi. The parametric curve is described as a locus, requiring the creation of a custom slider (at the top of the page).]

This course aims to provide the basic mathematical and physical concepts needed to create 3D representations, so that students can apply them in fields such as game simulation, scientific visualization, computer animation, and virtual object design.

The initial topics are dedicated to introducing the basic elements that will be worked with later: points and vectors. We will learn to work with these objects to calculate distances between points and angles between vectors.

Once the essential elements are defined, we will create derived objects such as lines and planes. These form the foundation of 3D object creation. We will learn to visualize these objects based on the camera's position.

Finally, we will study the essential equations to describe object motion. Through numerical integration of the motion equations, we will learn to move objects in a world with and without gravity, as well as bounce off walls or the ground.

In the first part of the course, we will study the mathematical concepts necessary for drawing, positioning, and orienting polygonal objects on the computer. These tools will allow us to draw simple objects and position them in 2D and 3D spaces. In the second part, we will study the essential physical laws that allow us to make objects move in these spaces.

Vector Spaces: Properties of vector spaces. Dot product. Linear combinations and basis. Vectors in 3 dimensions.

Matrices and Cross Product: Introduction to matrices. Identity and inverse matrices. Cross product. Solving systems of equations using matrices.

Transformations: Transformations in the plane. 3D transformations. Rotations around any axis. Homogeneous coordinates.

Equations of the Line: Lines in the plane. Distances. Relative position between lines. Geometric loci. Lines in 3D.

Equations of the Plane: Planes in 3D space. Intersection of lines and planes. Intersection of planes. Distance from a point to a plane. Projection onto the viewing plane.

The classes will alternate different methodologies:

- Theory classes where the general concepts of the different topics will be introduced

- Self-corrected questionnaires using the Moodle platform

- Practices writting short programs applying the concepts introduced in theory classes.

- Reading of didactic material where the physical and mathematical concepts are used to draw and move objects in virtual environments.

Annotation: Within the schedule set by the centre or degree programme, 15 minutes of one class will be reserved for students to evaluate their lecturers and their courses or modules through questionnaires.

If a student has failed any of the partial exams, they will have the option to take a resit exam for the specific partial they failed. These exams will be the same as those used in the continuous assessment as well as in the regular evaluation.

Once the recovery activities have been completed, the final grade for the subject will be determined by replacing the failed grades with the grades obtained in this phase, using the same weighting as inthe regular phase.

Use of Geogebra Software and an algebraic manipulation program (Maxima) to treat analysis problems, algebra and geometry. Particular attention is given to the consolidation, through the development and analysis of algorithms and geometric interpretation, of the concepts and problems covered in the courses Linear Algebra and Analytic Geometry I (M1010), Real Analysis I (M1011) and topics of Elementary Mathematics (M1024).

It is intended that at the end of the course, the student is capable of using Geogebra and a manipulation algebraic language (Maxima), dealing with problem of analysis, algebra and geometry, solving them, graphing and interpreting their solutions.

Introduction to Maxima:graphic interface; variables; functions; programming structure; graphic sketch.

Real functions of a real variable: sketch of the graph and interpretation; definition of the derivative function, tangent line of a curveat a point; calculation and geometric interpretation of limits; integral calculus and geometric interpretation; determination of maximum and minimum of functions. Limits of sequences.Approximate calculation of series sums. Polynomial approximation of functions.

Systems of linear equations: numerical resolution, graphical representation and interpretation of the solution; implementation in Maxima of Gauss Elimination Method and geometric interpretation. Spaces and vector subspaces: geometric representation and interpretation of linear combinations, subspaces generated by linear combinations of elements of a set, the sum of linear subspaces, bases. Linear maps: representation of the images of R2 and R3 subsets; invariant subsets and subspaces; calculation and geometric interpretation of the determinant and properties of a matrix of a linear application. Calculation and geometric interpretation of the internal product and norm of vectors, and the vector product in RR^3.

The study of revolution curves and solids may also be addressed.

Laboratory classes:resolution, by the teacher and the students, of exercises proposed in exercise sheets and / or proposed in class. Availability of slides to support classes; in particular supporting Maxima and solving some of the proposed exercises. Support to students in clarifying doubts in the contents and /or in solving exercises.

Students who have passed the course by taking the tests, and have not obtained the desired result, may take the exam in the normal period. In this case, students will have to choose, at the time of delivery of the exam, to waive or not the classification obtained in the assessment by tests (checking the desired option on the exam).

Students with a classification higher than or equal to 17.5 points (obtained in tests or in the exam of any of the seasons) may have to carry out a work in Maxima or a computer test with a written or oral component, to obtain a grade greater than or equal to 18 values.

En resumen, los diagramas fasoriales son una proyeccin de un vector giratorio sobre un eje horizontal que representa el valor instantneo. Puesto que se puede dibujar un diagrama fasorial para mostrar cualquier punto en el tiempo y, por lo tanto, cualquier ngulo, el vector de referencia de una cantidad alterna, tensin o corriente en el caso de circuitos elctricos, siempre se dibuja a lo largo de la direccin positiva del eje x.

Se recomienda siempre tomar como referencia la seal, tensin o corriente, comn en todos los elementos, por ejemplo, en caso de los elementos conectados en serie se recomienda tomar como referencia la seal de corriente, mientras los que estn conectados en paralelo se recomienda tomar la seal de tensin.

La seal de referencia se dibuja a lo largo del eje horizontal, y a partir de esta se trazan las otras seales, teniendo en cuenta las relaciones circuitales que caracterizan el circuito como las leyes de Kirchhoff de corriente y tensin las cuales deben cumplirse siempre, en ese sentido es importante conocer la suma y resta de vectores.

Las curvas sinusales con diferentes frecuencias no se pueden mostrar en el mismo diagrama vectorial debido a la diferente velocidad de los vectores. En cualquier momento, el ngulo de fase entre ellos es diferente.

Con el implemento de las TAC se potencia las nuevas posibilidades que ofrece la educacin a distancia, adems de la verificacin de los resultados tericos con los prcticos potenciando la autoevaluacin en el estudiante.

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