Using the [url=https://wiki.geogebra.org/en/3D_Graphics_Tools]construction [i]Tools[/i][/url] available in the [url=https://wiki.geogebra.org/en/3D_Graphics_View#3D_Graphics_View_Toolbar][i]3D Graphics View Toolbar[/i][/url] you can create geometric constructions in the [img width=16,height=16]https://wiki.geogebra.org/uploads/thumb/b/bb/Perspectives_algebra_3Dgraphics.svg/16px-Perspectives_algebra_3Dgraphics.svg.png[/img] [url=https://wiki.geogebra.org/en/3D_Graphics_View][i]3D Graphics View[/i] [/url]with the mouse. Select any construction tool from the [i]3D Graphics View Toolbar[/i] and read the tooltip provided in the [img width=16,height=16]https://wiki.geogebra.org/uploads/thumb/b/bb/Perspectives_algebra_3Dgraphics.svg/16px-Perspectives_algebra_3Dgraphics.svg.png[/img] [i]3D Graphics View[/i] in order to find out how to use the selected [url=https://wiki.geogebra.org/en/3D_Graphics_Tools][i]Tool[/i][/url]. [br] [b]Note:[/b] Any object you create in the [img width=16,height=16]https://wiki.geogebra.org/uploads/thumb/b/bb/Perspectives_algebra_3Dgraphics.svg/16px-Perspectives_algebra_3Dgraphics.svg.png[/img] [i]3D Graphics View[/i] also has an algebraic representation in the [img width=16,height=16]https://wiki.geogebra.org/uploads/thumb/4/40/Menu_view_algebra.svg/16px-Menu_view_algebra.svg.png[/img] [i][url=https://wiki.geogebra.org/en/Algebra_View]Algebra View[/url][/i] and vice versa any analytic representation in the [img width=16,height=16]https://wiki.geogebra.org/uploads/thumb/4/40/Menu_view_algebra.svg/16px-Menu_view_algebra.svg.png[/img] [i][url=https://wiki.geogebra.org/en/Algebra_View]Algebra View[/url][/i] has an graphical output in the [img width=16,height=16]https://wiki.geogebra.org/uploads/thumb/b/bb/Perspectives_algebra_3Dgraphics.svg/16px-Perspectives_algebra_3Dgraphics.svg.png[/img] [i]3D Graphics View[/i].[br][br][b][color=#1e84cc][size=150]Direct Input using the Input Bar[/size][/color][/b][br]GeoGebra’s [img width=16,height=16]https://wiki.geogebra.org/uploads/thumb/b/bb/Perspectives_algebra_3Dgraphics.svg/16px-Perspectives_algebra_3Dgraphics.svg.png[/img] [i]3D Graphics View[/i] supports points, vectors, lines, planes, and surfaces in a three-dimensional coordinate system. You may either use the [i][url=https://wiki.geogebra.org/en/Tools]Tools[/url][/i] provided in the [url=https://wiki.geogebra.org/en/3D_Graphics_View#3D_Graphics_View_Toolbar][i]3D Graphics View Toolbar[/i][/url], or directly enter the algebraic representation of these objects in the [i][url=https://wiki.geogebra.org/en/Input_Bar]Input Bar[/url][/i] or [url=https://wiki.geogebra.org/en/Input_Bar][i]Input Field[/i][/url] of the [img width=16,height=16]https://wiki.geogebra.org/uploads/thumb/4/40/Menu_view_algebra.svg/16px-Menu_view_algebra.svg.png[/img] [i][url=https://wiki.geogebra.org/en/Algebra_View]Algebra View[/url][/i] (GeoGebra Web and Tablet Apps).[br][b]Example:[/b] Enter [code]A=(5, -2, 1)[/code] into the [i]Input Bar[/i] or [i]Input Field[/i] of the [img width=16,height=16]https://wiki.geogebra.org/uploads/thumb/4/40/Menu_view_algebra.svg/16px-Menu_view_algebra.svg.png[/img] [i]Algebra View[/i] in order to create a point in the three-dimensional coordinate system.[br][b]Example:[/b] Enter [code]z = - x - y[/code] in order to create the corresponding surface.[br][br]Furthermore, you may now create curves, planes, as well as arbitrary parametric surfaces.[br]Command [url=https://wiki.geogebra.org/en/Surface_Command][code]Surface[/code][/url][code](x(s,t),y(s,t),z(s,t),s,0,1,t,-1,2)[br][/code] create surface given by parametric notation X(s,t) = [x(s,t),y(s,t),z(s,t)]. Bounds (0, 1) and (-1, 2) determine the size of rendered part of surface. [br]Tool [i]Rotate 3D Graphic View[/i] [icon]https://www.geogebra.org/images/ggb/toolbar/mode_rotateview.png[/icon] helps you to set up proper view direction.[br][br][b]Exercise:[/b][br]Line in 3D space is represented parametrically, by one-parameter vector function X(t) = [x(t), y(t), z(t)].[br]Is the given line parallel to any coordinate axis or plane (x-axis, y-axis, z-axis, xy-plane, yz-plane, xy-plane)?[br][list=1][*] X(t) = (1+t, 2+t, 1+t)[br]Rewrite the vector in algebraic window. Investigate the position of line in the [img width=16,height=16]https://wiki.geogebra.org/uploads/thumb/b/bb/Perspectives_algebra_3Dgraphics.svg/16px-Perspectives_algebra_3Dgraphics.svg.png[/img] [i]3D Graphics View[/i].[br]Solution is evident from the parametric representation. Line is given by point A = (1,2,1) and vector [i]u[/i] = (1,1,1). This vector is not parallel with any coordinate planes.[br][/*][*]Y(t) = (1 - t, 2 + t, 1)[br]Direction vector [i]v[/i] = (-1, 1, 0) is parallel with coordinate plane ([i]xy[/i]).[br][/*][/list]

[color=#0B5394][size=150]Task 2 - Geotest 11103[/size][/color][br]Choose the proper entity (shape) in [b] 3D [/b] which is expressed by the given equation or the parametric form.[br][list=1][*] a(s,t) = (1 + 2s - t, 2 - s, 1)[br]Write command [code]Surface(1 + 2s - t, 2 - s, 1, s, -2, 3, t ,-5, 5)[/code] into the [i]Input Bar[/i] of the [img width=16,height=16]https://wiki.geogebra.org/uploads/thumb/4/40/Menu_view_algebra.svg/16px-Menu_view_algebra.svg.png[/img] [i]Algebra View[/i]. Bounds (-2, 3) and (-5, 5) and determine only the size of rendered part of plane. You can change them later in order to fit the illustrativ view. [br]Solution is evident from the parametric representation. All coordinates are linear function of two parameters. Two parametric linear object is plane. Missing parameter gives us reference about parallelity with coordinate axis. For instance, function for third coordinate z=1 doesn't contain both parameters [i]s, t[/i]. That means, that plane [i]a[/i] must be parallel with [i]xy-plane[/i].[br][/*][*] x - y + 4 = 0[br]Enter [code]x - y + 4 = 0[/code] into the [i]Input Bar[/i] of the [img width=16,height=16]https://wiki.geogebra.org/uploads/thumb/4/40/Menu_view_algebra.svg/16px-Menu_view_algebra.svg.png[/img] [i]Algebra View[/i]. Object is depicted as line in pure Graphis window - intersection of plane and coordinate plane [i]xy[/i]. [img width=16,height=16]https://wiki.geogebra.org/uploads/thumb/b/bb/Perspectives_algebra_3Dgraphics.svg/16px-Perspectives_algebra_3Dgraphics.svg.png[/img] [i]3D Graphics View[/i] gives the whole plane. [br]Linear equation F(x, y, z) = 0 always represents plane in 3D space. Missing parameter gives us reference about parallelity with coordinate axis. General equation [i]x - y + 4 = 0[/i] doesn't contain [i]z[/i]. From that, plane is parallel with [i]z[/i]-axis.[br][/*][/list]