复数乘法的几何意义
2018-05-16 03:39
本范例直观展示了复数乘法的几何意义。在复平面中，两个复数相乘，其结果的模长等于两复数模长之积，辐角等于两复数辐角之和。通过构造相似三角形、线段和角度，该模型动态演示了复数乘法所对应的旋转与缩放变换，帮助学习者深入理解复数代数运算背后的几何直观与变换本质。
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.464.0" app="classic" platform="w" id="9dc8b91d-99ee-47d4-a341-a7fd993d5b7a" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1019" height="687" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.7841020608439647" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1,1" size="400" window="924,277,700,631" />
<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="false" location="1,1,1" size="500" window="100,100,600,400" />
<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="false" location="3,3" size="303" 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="1" visible="true" inframe="false" stylebar="false" location="3" size="799" window="100,100,600,400" />
<view id="16" visible="true" inframe="false" stylebar="false" location="1" size="210" 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="128" visible="false" inframe="true" stylebar="false" location="1" size="480" 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" />
<view id="43" visible="false" inframe="false" stylebar="false" location="1" size="450" window="50,50,500,500" />
</views>
<toolbar show="false" 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="false" cmd="true" top="algebra" />
<dockBar show="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="2"/>
<font size="24"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="798" height="684"/>
<coordSystem xZero="385.4641720038519" yZero="349.116282318238" scale="71.334395605562" yscale="71.33439560556214"/>
<evSettings axes="true" grid="false" 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="1" grid="0"/>
<axis id="0" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<grid distX="1" distY="1" distTheta="0.5235987755982988"/>
</euclidianView>
<euclidianView>
<viewNumber viewNo="2"/>
<size width="209" height="684"/>
<coordSystem xZero="207.58179784664458" yZero="315.0957165742532" scale="37.73668744683365" yscale="37.736687446833656"/>
<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>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<element type="boolean" label="b">
<value val="true"/>
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="7" y="64"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<ggbscript onUpdate="If(b,SetValue[c,false],SetValue[c,true]) SetValue[k,0] SetValue[p,false] SetValue[j,false]"/>
<caption val="乘法"/>
</element>
<element type="numeric" label="k">
<value val="0"/>
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelOffset x="35" y="0"/>
<labelMode val="1"/>
<slider min="0" max="10" absoluteScreenLocation="true" width="150" x="22" y="204" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation speed="5" type="3" playing="false"/>
</element>
<expression label ="O" exp="(0, 0)" type="point"/>
<element type="point" label="O">
<show object="true" label="false" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="0" y="0" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<element type="point" label="Z_1">
<show object="true" label="false" ev="4"/>
<objColor r="204" g="102" b="0" alpha="0"/>
<layer val="9"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="-1" y="-1" z="1"/>
<pointSize val="7"/>
<pointStyle val="0"/>
</element>
<command name="Angle">
<input a0="(2, 0)" a1="O" a2="Z_1"/>
<output a0="α"/>
</command>
<element type="angle" label="α">
<angleStyle val="0"/>
<value val="3.9269908169872414"/>
<show object="false" label="false" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="2"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="75"/>
</element>
<command name="Segment">
<input a0="O" a1="Z_1"/>
<output a0="u_3"/>
</command>
<element type="segment" label="u_3">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="1" y="-1" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="r_1" exp="1 + (k * (u_3 - 1) / 10)" />
<element type="numeric" label="r_1">
<value val="1"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<expression label="r_3" exp="(u_3 * (u_3 - ((k * (u_3 - 1))) / 10) / u_3)" />
<element type="numeric" label="r_3">
<value val="1.4142135623730951"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<expression label="r_3'" exp="r_3 / u_3" />
<element type="numeric" label="r_3'">
<value val="1"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<command name="If">
<input a0="b" a1="((r_1 * cos((k / 10 * α))), (r_1 * sin((k / 10 * α))))" a2="((r_3' * cos((((-k)) / 10 * α))), (r_3' * sin((((-k)) / 10 * α))))"/>
<output a0="U"/>
</command>
<element type="point" label="U">
<show object="false" label="false" ev="8"/>
<condition showObject="b"/>
<objColor r="0" g="255" b="255" alpha="0"/>
<layer val="8"/>
<labelMode val="0"/>
<coords x="1" y="0" z="1"/>
<pointSize val="6"/>
<pointStyle val="0"/>
</element>
<command name="Ray">
<input a0="O" a1="U"/>
<output a0="g"/>
</command>
<element type="ray" label="g">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="8"/>
<labelMode val="0"/>
<coords x="0" y="1" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Point">
<input a0="g"/>
<output a0="A"/>
</command>
<element type="point" label="A">
<show object="true" label="false" ev="4"/>
<objColor r="125" g="125" b="255" alpha="0"/>
<layer val="8"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="0" y="0" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Circle">
<input a0="A" a1="5"/>
<output a0="e"/>
</command>
<element type="conic" label="e">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="8"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1" A1="1" A2="-25" A3="0" A4="0" A5="0"/>
<eqnStyle style="specific"/>
</element>
<command name="Intersect">
<input a0="e" a1="g" a2="1"/>
<output a0="UL"/>
</command>
<element type="point" label="UL">
<show object="false" label="false" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="8"/>
<labelMode val="0"/>
<coords x="5" y="0" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<expression label ="X" exp="(1, 0)" type="point"/>
<element type="point" label="X">
<show object="false" label="false" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="1" y="0" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Angle">
<input a0="(5, 0)" a1="O" a2="UL"/>
<output a0="β"/>
</command>
<element type="angle" label="β">
<angleStyle val="0"/>
<value val="0"/>
<show object="false" label="false" ev="4"/>
<condition showObject="a ∧ b"/>
<objColor r="0" g="153" b="0" alpha="0.5"/>
<layer val="0"/>
<labelMode val="2"/>
<lineStyle thickness="7" type="0" typeHidden="2" opacity="178"/>
<arcSize val="90"/>
</element>
<command name="Segment">
<input a0="O" a1="U"/>
<output a0="u_1"/>
</command>
<element type="segment" label="u_1">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0" y="1" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="s_1" exp="tan(β)" />
<element type="numeric" label="s_1">
<value val="0"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<expression label="d_1" exp="(u_1 * (1 + s_1^(2))^(0.5))" />
<element type="numeric" label="d_1">
<value val="1"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<expression label="s_2" exp="-1 / s_1" />
<element type="numeric" label="s_2">
<value val="-Infinity"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<expression label="d_2" exp="(u_1 * (1 + s_2^(2))^(0.5))" />
<element type="numeric" label="d_2">
<value val="Infinity"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<element type="point" label="Z_2">
<show object="true" label="false" ev="4"/>
<condition showObject="p ∨ j"/>
<objColor r="0" g="153" b="204" alpha="0"/>
<layer val="9"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="1" y="2" z="1"/>
<pointSize val="7"/>
<pointStyle val="0"/>
</element>
<command name="Angle">
<input a0="X" a1="O" a2="Z_2"/>
<output a0="γ"/>
</command>
<element type="angle" label="γ">
<angleStyle val="0"/>
<value val="1.1071487177940904"/>
<show object="false" label="false" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="2"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="30"/>
</element>
<command name="Segment">
<input a0="Z_2" a1="O"/>
<output a0="u_2"/>
</command>
<element type="segment" label="u_2">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="2" y="-1" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="r_2" exp="(u_2 * (1 + (k * (u_3 - 1) / 10)))" />
<element type="numeric" label="r_2">
<value val="2.23606797749979"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<expression label="Z_2'" exp="((r_2 * cos((k * α / 10) + γ)), (r_2 * sin((k * α / 10) + γ)))" type="point" />
<element type="point" label="Z_2'">
<show object="false" label="false" ev="4"/>
<condition showObject="k > 0 ∧ b ∧ p"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="8"/>
<labelMode val="0"/>
<coords x="1.0000000000000002" y="2" z="1"/>
<pointSize val="7"/>
<pointStyle val="0"/>
</element>
<command name="Ray">
<input a0="O" a1="U"/>
<output a0="f"/>
</command>
<element type="ray" label="f">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0" y="1" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="Z_2'" a1="O"/>
<output a0="u_4"/>
</command>
<element type="segment" label="u_4">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="79" y="-15"/>
<labelMode val="0"/>
<coords x="2" y="-1.0000000000000002" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="boolean" label="a">
<value val="true"/>
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="9" y="226"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<caption val="角度"/>
</element>
<expression label="Z_1'" exp="((r_3 * cos(α - (k * α / 10))), (r_3 * sin(α - (k * α / 10))))" type="point" />
<element type="point" label="Z_1'">
<show object="true" label="false" ev="4"/>
<condition showObject="k > 0 ∧ c"/>
<objColor r="0" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-1.0000000000000002" y="-1" z="1"/>
<pointSize val="6"/>
<pointStyle val="0"/>
</element>
<element type="boolean" label="c">
<value val="false"/>
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="5" y="104"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<ggbscript onUpdate="If(c,SetValue[b,false],SetValue[b,true]) SetValue[k,0] SetValue[j,false] SetValue[p,false]"/>
<caption val="除法"/>
</element>
<expression label="U_q" exp="((r_3' * cos((((-k)) / 10 * α))), (r_3' * sin((((-k)) / 10 * α))))" type="point" />
<element type="point" label="U_q">
<show object="false" label="false" ev="8"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="1" y="0" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<expression label="textZ_2r" exp=""" + (LaTeX[1 + (k * (u_3 - 1) / 10)]) + """ />
<element type="text" label="textZ_2r">
<show object="true" label="true" ev="40"/>
<condition showObject="k > 0 ∧ p ∧ text ∧ b"/>
<objColor r="204" g="102" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<startPoint exp="Z_2' + (1.5, 0.5)"/>
<labelOffset x="-32" y="-27"/>
</element>
<expression label="textZ_2a'" exp=""" + (LaTeX[(k * α / 10)]) + """ />
<element type="text" label="textZ_2a'">
<show object="true" label="true" ev="8"/>
<condition showObject="k > 0 ∧ p ∧ text ∧ b"/>
<objColor r="204" g="102" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<startPoint exp="Z_2' + (8, 0.5)"/>
<labelOffset x="-40" y="-28"/>
</element>
<expression label="textZ_2a" exp=""" + (LaTeX[(k * α / 10)]) + """ />
<element type="text" label="textZ_2a">
<show object="true" label="true" ev="8"/>
<condition showObject="k > 0 ∧ p ∧ text ∧ b"/>
<objColor r="204" g="102" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<startPoint exp="Z_2' + (4, 0.5)"/>
<labelOffset x="-9" y="-28"/>
</element>
<element type="boolean" label="p">
<value val="true"/>
<show object="true" label="true" ev="3"/>
<condition showObject="b"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="98" y="67"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<caption val="被乘數"/>
</element>
<element type="boolean" label="d">
<value val="true"/>
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="10" y="270"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<caption val="虛擬格線"/>
</element>
<expression label="textZ_2" exp=""$" + (LaTeX[u_2]) + "" + (LaTeX[If[k > 0, "×\hspace{1.8em}", ""]]) + "\left(\cos(" + (LaTeX[γ]) + "" + (LaTeX[If[k > 0, "+\hspace{2.8em}", ""]]) + ")\,,\,\sin(" + (LaTeX[γ]) + "" + (LaTeX[If[k > 0, "+\hspace{2.8em}", ""]]) + ")\right)$"" />
<element type="text" label="textZ_2">
<show object="true" label="true" ev="8"/>
<condition showObject="p ∧ k > 0 ∧ text ∧ b"/>
<objColor r="0" g="153" b="204" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<startPoint exp="Z_2'"/>
<labelOffset x="9" y="-63"/>
</element>
<element type="boolean" label="text">
<value val="true"/>
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="93" y="228"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<caption val="文字"/>
</element>
<command name="Ray">
<input a0="A" a1="Z_2"/>
<output a0="h"/>
</command>
<element type="ray" label="h">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="8"/>
<labelMode val="0"/>
<coords x="-2" y="1" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Ray">
<input a0="A" a1="Z_2'"/>
<output a0="i"/>
</command>
<element type="ray" label="i">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="8"/>
<labelMode val="0"/>
<coords x="-2" y="1.0000000000000002" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Intersect">
<input a0="e" a1="i" a2="1"/>
<output a0="Z_2'L"/>
</command>
<element type="point" label="Z_2'L">
<show object="false" label="false" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="8"/>
<labelMode val="0"/>
<coords x="2.2360679774997902" y="4.47213595499958" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Intersect">
<input a0="e" a1="h" a2="1"/>
<output a0="Z_2L"/>
</command>
<element type="point" label="Z_2L">
<show object="false" label="false" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="8"/>
<labelMode val="0"/>
<coords x="2.23606797749979" y="4.47213595499958" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Angle">
<input a0="Z_2L" a1="O" a2="Z_2'L"/>
<output a0="δ"/>
</command>
<element type="angle" label="δ">
<angleStyle val="0"/>
<value val="0"/>
<show object="false" label="false" ev="4"/>
<condition showObject="a ∧ p ∧ b"/>
<objColor r="0" g="100" b="0" alpha="0.5"/>
<layer val="0"/>
<labelMode val="2"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="100"/>
</element>
<command name="Sequence">
<input a0="If[β ≠ π / 2 ∧ β ≠ (3 * π / 2), y - (s_1 * x) = (t * d_1), x = (t * u_1)]" a1="t" a2="-40" a3="40"/>
<output a0="list_1"/>
</command>
<element type="list" label="list_1">
<show object="true" label="false" ev="8"/>
<condition showObject="d"/>
<objColor r="224" g="224" b="224" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="-1"/>
</element>
<command name="Sequence">
<input a0="If[β ≠ π / 2 ∧ β ≠ (3 * π / 2) ∧ β ≠ 0, y - (s_2 * x) = (t * d_2), If[β ≠ 0, y = (t * u_1), x = (t * u_1)]]" a1="t" a2="-40" a3="40"/>
<output a0="list_2"/>
</command>
<element type="list" label="list_2">
<show object="true" label="false" ev="8"/>
<condition showObject="d"/>
<objColor r="224" g="224" b="224" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="-1"/>
</element>
<expression label="Z_2''" exp="(((r_3' * u_2) * cos(γ - (k / 10 * α))), ((r_3' * u_2) * sin(γ - (k / 10 * α))))" type="point" />
<element type="point" label="Z_2''">
<show object="true" label="false" ev="8"/>
<condition showObject="k > 0 ∧ c ∧ j"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="8"/>
<labelMode val="0"/>
<coords x="1.0000000000000002" y="2" z="1"/>
<pointSize val="7"/>
<pointStyle val="0"/>
</element>
<element type="boolean" label="j">
<value val="false"/>
<show object="true" label="true" ev="3"/>
<condition showObject="c"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="94" y="108"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<caption val="被除數"/>
</element>
<expression label="textZ_2'r" exp=""" + (LaTeX[1 / r_3']) + """ />
<element type="text" label="textZ_2'r">
<show object="true" label="true" ev="40"/>
<condition showObject="j ∧ text ∧ k > 0"/>
<objColor r="204" g="102" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<startPoint exp="Z_2''"/>
<labelOffset x="-8" y="46"/>
</element>
<expression label="textZ_2'a" exp=""" + (LaTeX[(k * α / 10)]) + """ />
<element type="text" label="textZ_2'a">
<show object="true" label="true" ev="40"/>
<condition showObject="j ∧ text ∧ k > 0"/>
<objColor r="204" g="102" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<startPoint exp="Z_2'' + (3, -0.5)"/>
<labelOffset x="-10" y="-14"/>
</element>
<expression label="textZ_2'a'" exp=""" + (LaTeX[(k * α / 10)]) + """ />
<element type="text" label="textZ_2'a'">
<show object="true" label="true" ev="40"/>
<condition showObject="j ∧ text ∧ k > 0"/>
<objColor r="204" g="102" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<startPoint exp="Z_2'' + (7, -0.5)"/>
<labelOffset x="-50" y="-13"/>
</element>
<expression label="text3" exp=""$\displaystyle\frac{\hspace{0.4em}" + (LaTeX[u_2]) + "\hspace{0.4em}}{}\left(\cos(" + (LaTeX[γ]) + "" + (LaTeX[If[k > 0, "-\hspace{2.9em}", ""]]) + "),\sin(" + (LaTeX[γ]) + "" + (LaTeX[If[k > 0, "-\hspace{2.9em}", ""]]) + ")\right)$"" />
<element type="text" label="text3">
<show object="true" label="false" ev="8"/>
<condition showObject="j ∧ text ∧ k > 0"/>
<objColor r="0" g="153" b="204" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<startPoint exp="Z_2'' + (-0.5, -1.2)"/>
<labelOffset x="21" y="-77"/>
</element>
<command name="Ray">
<input a0="A" a1="Z_1'"/>
<output a0="l"/>
</command>
<element type="ray" label="l">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="8"/>
<labelMode val="0"/>
<coords x="1" y="-1.0000000000000002" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Ray">
<input a0="A" a1="Z_1"/>
<output a0="m"/>
</command>
<element type="ray" label="m">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="8"/>
<labelMode val="0"/>
<coords x="1" y="-1" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Ray">
<input a0="A" a1="Z_2''"/>
<output a0="n"/>
</command>
<element type="ray" label="n">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="8"/>
<labelMode val="0"/>
<coords x="-2" y="1.0000000000000002" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Ray">
<input a0="A" a1="Z_2"/>
<output a0="q"/>
</command>
<element type="ray" label="q">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="8"/>
<labelMode val="0"/>
<coords x="-2" y="1" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Intersect">
<input a0="e" a1="l" a2="1"/>
<output a0="Z_1'L"/>
</command>
<element type="point" label="Z_1'L">
<show object="false" label="false" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="8"/>
<labelMode val="0"/>
<coords x="-3.5355339059327378" y="-3.535533905932737" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Intersect">
<input a0="e" a1="m" a2="1"/>
<output a0="Z_1L"/>
</command>
<element type="point" label="Z_1L">
<show object="false" label="false" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="8"/>
<labelMode val="0"/>
<coords x="-3.5355339059327373" y="-3.5355339059327373" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Intersect">
<input a0="e" a1="n" a2="1"/>
<output a0="Z_2''L"/>
</command>
<element type="point" label="Z_2''L">
<show object="false" label="false" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="8"/>
<labelMode val="0"/>
<coords x="2.2360679774997902" y="4.47213595499958" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Angle">
<input a0="Z_1'L" a1="A" a2="Z_1L"/>
<output a0="ζ"/>
</command>
<element type="angle" label="ζ">
<angleStyle val="0"/>
<value val="0"/>
<show object="true" label="false" ev="4"/>
<condition showObject="c ∧ a"/>
<objColor r="0" g="153" b="0" alpha="0.44999998807907104"/>
<layer val="8"/>
<labelMode val="2"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="95"/>
</element>
<command name="Ray">
<input a0="A" a1="U_q"/>
<output a0="r"/>
</command>
<element type="ray" label="r">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="8"/>
<labelMode val="0"/>
<coords x="0" y="1" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="text1" exp=""$" + (LaTeX[If[b, "乘數", "除數"]]) + "\\" + (LaTeX[u_3]) + "\left(\cos" + (LaTeX[α]) + ",\sin" + (LaTeX[α]) + "\right)$"" />
<element type="text" label="text1">
<show object="true" label="false" ev="8"/>
<condition showObject="text"/>
<objColor r="204" g="102" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<startPoint exp="Z_1"/>
<labelOffset x="-22" y="-77"/>
</element>
<expression label="text2" exp=""$被" + (LaTeX[If[b, "乘", "除"]]) + "數\\ \displaystyle" + (LaTeX[u_2]) + "\left(\cos(" + (LaTeX[γ]) + "),\sin(" + (LaTeX[γ]) + ")\right)$"" />
<element type="text" label="text2">
<show object="true" label="false" ev="8"/>
<condition showObject="(p ∨ j) ∧ text"/>
<objColor r="0" g="153" b="204" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<startPoint exp="Z_2"/>
<labelOffset x="-4" y="-80"/>
</element>
<command name="Angle">
<input a0="(5, 0)" a1="O" a2="Z_2L"/>
<output a0="ε"/>
</command>
<element type="angle" label="ε">
<angleStyle val="0"/>
<value val="1.1071487177940904"/>
<show object="true" label="false" ev="4"/>
<condition showObject="a ∧ (p ∨ j)"/>
<objColor r="0" g="0" b="255" alpha="0.6000000238418579"/>
<layer val="0"/>
<labelMode val="2"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="75"/>
</element>
<command name="Angle">
<input a0="Z_2''L" a1="A" a2="Z_2L"/>
<output a0="η"/>
</command>
<element type="angle" label="η">
<angleStyle val="0"/>
<value val="0"/>
<show object="true" label="false" ev="4"/>
<condition showObject="c ∧ a"/>
<objColor r="0" g="100" b="0" alpha="0.5"/>
<layer val="8"/>
<labelMode val="2"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="100"/>
</element>
</construction>
</geogebra>