Construction and Demonstration of Vector Arrays
2020-10-09 17:45
This example demonstrates how to construct 2D vector arrays using the Flatten and Sequence commands in GeoGebra. By defining base points and vectors, combined with sequence and list operations, a regularly arranged set of vectors is generated. This demonstration helps in understanding the geometric interpretation of vectors, linear combinations, and the application of geometric transformations in
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.608.0" app="classic" platform="d" id="c49e0e97-b6d9-4274-8b7a-534794180f36" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/apps/xsd/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1715" height="658" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.5393586005830904" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1,1" size="400" window="700,407,1159,747" />
<view id="4" toolbar="0 || 2020 , 2021 , 2022 , 66 || 2001 , 2003 , 2002 , 2004 , 2005 || 2040 , 2041 , 2042 , 2044 , 2043" visible="false" inframe="false" stylebar="false" location="1,1" size="300" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="true" location="3,3" size="357" tab="ALGEBRA" window="100,100,600,400" />
<view id="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 || 1008 1009 || 66 68 || 6" visible="false" inframe="false" stylebar="false" location="1,3" size="300" window="100,100,600,400" />
<view id="512" toolbar="0 | 1 501 5 19 , 67 | 2 15 45 18 , 7 37 | 514 3 9 , 13 44 , 47 | 16 | 551 550 11 , 20 22 21 23 , 55 56 57 , 12 | 69 | 510 511 , 512 513 | 533 531 , 534 532 , 522 523 , 537 536 , 535 | 521 520 | 36 , 38 49 560 | 571 30 29 570 31 33 | 17 | 540 40 41 42 , 27 28 35 , 6 , 502" visible="false" inframe="false" stylebar="true" location="1,1" size="432" window="100,100,600,400" />
<view id="1" visible="true" inframe="false" stylebar="true" location="3" size="925" window="100,100,600,400" />
<view id="16" visible="true" inframe="false" stylebar="false" location="1" size="752" window="50,50,500,500" />
<view id="32" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="64" toolbar="0" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
</views>
<toolbar show="true" items="0 39 73 62 | 1 501 67 , 5 19 , 72 75 76 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 | 16 51 64 , 70 | 10 34 53 11 , 24 20 22 , 21 23 | 55 56 57 , 12 | 36 46 , 38 49 50 , 71 14 68 | 30 29 54 32 31 33 | 25 17 26 60 52 61 | 40 41 42 , 27 28 35 , 6" position="1" help="false" />
<input show="true" cmd="true" top="algebra" />
<dockBar show="true" east="true" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="18"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="925" height="409"/>
<coordSystem xZero="49.49999999999974" yZero="289.0" scale="49.99999999999998" yscale="49.99999999999998"/>
<evSettings axes="true" grid="true" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="0"/>
<bgColor r="255" g="255" b="255"/>
<axesColor r="0" g="0" b="0"/>
<gridColor r="192" g="192" b="192"/>
<lineStyle axes="11" grid="0"/>
<labelStyle axes="1" serif="true"/>
<axis id="0" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<euclidianView>
<viewNumber viewNo="2"/>
<size width="752" height="442"/>
<coordSystem xZero="6.00000000000002" yZero="504.99999999999966" scale="49.99999999999994" yscale="49.99999999999995"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="3"/>
<bgColor r="255" g="255" b="255"/>
<axesColor r="0" g="0" b="0"/>
<gridColor r="192" g="192" b="192"/>
<lineStyle axes="1" grid="0"/>
<axis id="0" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<algebraView>
<mode val="1"/>
<auxiliary show="true"/>
</algebraView>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<tableview min="-2.0" max="2.0" step="1.0"/>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<expression label="A" exp="(0, 0)" type="point"/>
<element type="point" label="A">
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="0.0" y="0.0" z="1.0"/>
<pointSize val="1"/>
<pointStyle val="10"/>
</element>
<expression label="B" exp="(1, 1)" type="point"/>
<element type="point" label="B">
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="1.0" y="1.0" z="1.0"/>
<pointSize val="1"/>
<pointStyle val="10"/>
</element>
<expression label="C" exp="(2, 0)" type="point"/>
<element type="point" label="C">
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="2.0" y="0.0" z="1.0"/>
<pointSize val="1"/>
<pointStyle val="10"/>
</element>
<expression label="D" exp="(3, 2)" type="point"/>
<element type="point" label="D">
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="3.0" y="2.0" z="1.0"/>
<pointSize val="1"/>
<pointStyle val="10"/>
</element>
<expression label="l1" exp="{A, B, C, D}" />
<element type="list" label="l1">
<show object="false" label="false"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<element type="numeric" label="n">
<value val="3.0"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="30" absoluteScreenLocation="true" width="200.0" x="199.0" y="139.0" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
<ggbscript onUpdate=""/>
</element>
<expression label="u" exp="(4, 0)" type="vector"/>
<element type="vector" label="u">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<coords x="4.0" y="0.0" z="0.0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
</element>
<command name="Flatten">
<input a0="Sequence[Translate[l1, Vector[(k * u)]], k, 0, n]"/>
<output a0="l2"/>
</command>
<element type="list" label="l2">
<show object="true" label="false"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Vector[l2(k), l2(k + 1)]" a1="k" a2="1" a3="Length[l2]"/>
<output a0="l3"/>
</command>
<element type="list" label="l3">
<show object="true" label="false"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<expression label="text1" exp=""\Large{主题：点阵制作}\\
1. 周期为4的点列，先做出第一个周期的四个点A、B、C、D\\
2. 四点打包成一个列表 list1={A,B,C,D} 按顺序放置\\
3. 制作一个平移向量 u = (4,0), 周期为4的缘故\\
4. 制作一个控制平移次数的滑块 n=Slider(0,30,1)\\
5. 巧妙利用序列命令产生系列点阵 l2=Sequence(Translate(l1, k*u),k=0,n)\\
6. 上述产生的点列是二维的，需要扁平化 l2=Flatten(l2)\\
7. 构造向量阵列 l3=Sequence(Vector(l2(k),l2(k+1)),k,0,length(l2))\\
 这里的 l2(k) 表示 列表l2中的第k个元素（点），\\
 也可以改成不会产生歧义的的命令 Element(l2,k)\\
8. 改变滑动条n，即可看到动态生成效果。\\
9. 增加点坐标阵列显示 l3=Sequence(Text(l2(k), l2(k)), k, 0, Length(l2))""/>
<element type="text" label="text1">
<show object="true" label="false" ev="43"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<bgColor r="255" g="192" b="203" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.0" size="0" style="0"/>
<auxiliary val="true"/>
<startPoint x="0.07000000000000384" y="9.720000000000006" z="1.0"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
</element>
<expression label="text2" exp=""题目：第2020个点的坐标是多少？""/>
<element type="text" label="text2">
<show object="true" label="false" ev="40"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<bgColor r="208" g="240" b="192" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<font serif="false" sizeM="1.4" size="8" style="1"/>
<absoluteScreenLocation x="153" y="74"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
</element>
<command name="Sequence">
<input a0="Text[l2(k), l2(k)]" a1="k" a2="0" a3="Length[l2]"/>
<output a0="l4"/>
</command>
<element type="list" label="l4">
<show object="true" label="false"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
</construction>
</geogebra>