线性组合与三角形：向量交互式练习
2016-12-11 12:35
本GeoGebra范例演示向量的线性组合如何构造三角形。通过平移(Translate)、旋转(Rotate)和向量(Vector)运算，展示两个基向量u、v的线性组合生成三角形顶点的几何过程。包含随机出题功能(Random)，支持交互式练习向量加法与标量乘法在平面几何中的应用，帮助理解线性组合的几何意义。
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.303.0" id="f7f04227-ff47-423a-b914-4f720121b5e6" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1000" height="500" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.562" orientation="1" />
</panes>
<views>
<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="323" window="100,100,250,400" />
<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="4097" visible="false" inframe="false" stylebar="true" location="1,1" size="141" window="-2546,28,1266,768" />
<view id="1" visible="true" inframe="false" stylebar="true" location="3" size="562" window="100,100,600,400" />
<view id="16" visible="true" inframe="false" stylebar="true" location="1" size="428" 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" />
<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="true" stylebar="false" location="1" size="480" window="50,50,500,500" />
</views>
<toolbar show="false" items="0 39 | 1 501 67 , 5 19 , 72 | 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 | 30 29 54 32 31 33 | 17 26 62 73 , 14 68 | 25 52 60 61 | 40 41 42 , 27 28 35 , 6" position="1" help="false" />
<input show="false" cmd="true" top="false" />
<dockBar show="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="2"/>
<font size="28"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="562" height="498"/>
<coordSystem xZero="274.27798849403683" yZero="292.5388999753613" scale="60.92880457409231" yscale="60.92880457409246"/>
<evSettings axes="false" 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="10"/>
<axis id="0" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<euclidianView>
<viewNumber viewNo="2"/>
<size width="428" height="498"/>
<coordSystem xZero="142.99999999999994" yZero="451.9999999999998" scale="49.999999999999986" yscale="49.99999999999995"/>
<evSettings axes="false" 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="10"/>
<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="">
<command name="Intersect">
<input a0="xAxis" a1="yAxis"/>
<output a0="A"/>
</command>
<element type="point" label="A">
<show object="true" label="false" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0" y="0" z="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Point">
<input a0="xAxis"/>
<output a0="B"/>
</command>
<element type="point" label="B">
<show object="false" label="false" ev="4"/>
<objColor r="125" g="125" 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="3"/>
<pointStyle val="0"/>
</element>
<command name="Polygon">
<input a0="A" a1="B" a2="3"/>
<output a0="poly1" a1="b" a2="c" a3="d" a4="C"/>
</command>
<element type="polygon" label="poly1">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="false" label="false" ev="4"/>
<objColor r="153" g="51" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="b">
<show object="true" label="false" ev="4"/>
<objColor r="153" g="51" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="0" y="1" z="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="c">
<show object="true" label="false" ev="4"/>
<objColor r="153" g="51" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<coords x="-0.8660254037844388" y="-0.4999999999999998" z="0.8660254037844388"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="d">
<show object="true" label="false" ev="4"/>
<objColor r="153" g="51" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<coords x="0.8660254037844388" y="-0.5000000000000002" z="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="point" label="C">
<show object="true" label="false" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<coords x="0.5000000000000002" y="0.8660254037844388" z="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Translate">
<input a0="poly1" a1="(1.5, 0.8660254037844386)"/>
<output a0="poly1'_2"/>
</command>
<element type="polygon" label="poly1'_2">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="false" label="false" ev="4"/>
<objColor r="153" g="51" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<command name="Translate">
<input a0="poly1" a1="(1.5, -0.8660254037844386)"/>
<output a0="poly1'"/>
</command>
<element type="polygon" label="poly1'">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="false" label="false" ev="4"/>
<objColor r="153" g="51" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="point" label="O">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="-30" y="35"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="0" y="0" z="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<element type="point" label="U">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" 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="3"/>
<pointStyle val="0"/>
</element>
<element type="point" label="V">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="-0.496940465280545" y="0.8622998652709707" z="1"/>
<coordStyle style="polar"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Vector">
<input a0="O" a1="U"/>
<output a0="u"/>
</command>
<element type="vector" label="u">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="1" y="0" z="0"/>
<lineStyle thickness="7" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
<startPoint exp="O"/>
</element>
<command name="Vector">
<input a0="O" a1="V"/>
<output a0="v"/>
</command>
<element type="vector" label="v">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.496940465280545" y="0.8622998652709707" z="0"/>
<lineStyle thickness="7" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
<startPoint exp="O"/>
</element>
<element type="numeric" label="px">
<value val="1"/>
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelOffset x="19" y="7"/>
<labelMode val="2"/>
<slider min="-3" max="3" absoluteScreenLocation="true" width="200" x="43" y="202" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="py">
<value val="-1"/>
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelOffset x="36" y="-2"/>
<labelMode val="2"/>
<slider min="-3" max="3" absoluteScreenLocation="true" width="200" x="43" y="252" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.5" speed="1" type="0" playing="false"/>
</element>
<command name="Vector">
<input a0="O" a1="(px * U) + (py * V)"/>
<output a0="p"/>
</command>
<element type="vector" label="p">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="8" y="10"/>
<labelMode val="0"/>
<coords x="1.496940465280545" y="-0.8622998652709707" z="0"/>
<lineStyle thickness="7" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
<startPoint exp="O"/>
</element>
<command name="Random">
<input a0="-3" a1="3"/>
<output a0="qx"/>
</command>
<element type="numeric" label="qx">
<value val="-1"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<command name="Random">
<input a0="-2" a1="3"/>
<output a0="qy"/>
</command>
<element type="numeric" label="qy">
<value val="-1"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="Q" exp="(qx * U) + (qy * V)" type="point" />
<element type="point" label="Q">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="-37" y="6"/>
<labelMode val="0"/>
<coords x="-0.5030595347194551" y="-0.8622998652709707" z="1"/>
<pointSize val="5"/>
<pointStyle val="4"/>
</element>
<element type="button" label="重新出題">
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="44" y="14"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<ggbscript val="qx=Random[-3,3] qy=Random[-2,3] a=false"/>
<font serif="false" sizeM="2" size="28" style="0"/>
</element>
<expression label="text2" exp=""$x$""/>
<element type="text" label="text2">
<show object="true" label="false" ev="43"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<startPoint x="-2.64" y="5.740000000000001" z="1"/>
</element>
<expression label="text2_1" exp=""$y$""/>
<element type="text" label="text2_1">
<show object="true" label="false" ev="43"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<startPoint x="-2.66" y="4.860000000000001" z="1"/>
</element>
<expression label="text3" exp=""正確""/>
<element type="text" label="text3">
<show object="true" label="false" ev="43"/>
<condition showObject="qx ≟ px ∧ qy ≟ py"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="false" sizeM="2" size="28" style="0"/>
<startPoint x="2.6400000000000023" y="4.420000000000001" z="1"/>
</element>
<element type="numeric" label="size">
<value val="3"/>
<show object="false" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="2"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<command name="Sequence">
<input a0="Translate[Line[O, V], Vector[(i * u)]]" a1="i" a2="(-2 * size)" a3="(2 * size)"/>
<output a0="yax"/>
</command>
<element type="list" label="yax">
<show object="false" label="false" ev="4"/>
<condition showObject="a"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="20" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<command name="Sequence">
<input a0="Translate[Line[O, U], Vector[(i * v)]]" a1="i" a2="(-2 * size)" a3="(2 * size)"/>
<output a0="list1"/>
</command>
<element type="list" label="list1">
<show object="true" label="false" ev="4"/>
<condition showObject="a"/>
<objColor r="255" g="0" b="255" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="20" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<element type="boolean" label="a">
<value val="false"/>
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="51" y="318"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<caption val="輔助線"/>
</element>
<expression label="text4" exp=""$\overset{\rightharpoonup}{u}$""/>
<element type="text" label="text4">
<show object="true" label="false" ev="40"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<startPoint exp="(O + U) / 2"/>
</element>
<expression label="text5" exp=""$\overset{\rightharpoonup}{v}$""/>
<element type="text" label="text5">
<show object="true" label="false" ev="40"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<startPoint exp="(O + V) / 2"/>
<labelOffset x="-53" y="-35"/>
</element>
<command name="Sequence">
<input a0="Sequence[Translate[poly1, Vector[(i * (1, 0)) + (j * (0.5, ((-sqrt(3))) / 2))]], i, (-size), size]" a1="j" a2="(-size)" a3="size"/>
<output a0="list4"/>
</command>
<element type="list" label="list4">
<show object="true" label="false" ev="8"/>
<objColor r="0" g="100" b="0" alpha="0.1"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<command name="Rotate">
<input a0="poly1" a1="60°" a2="O"/>
<output a0="poly2"/>
</command>
<element type="polygon" label="poly2">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false" ev="4"/>
<objColor r="153" g="51" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<command name="Sequence">
<input a0="Sequence[Translate[poly2, Vector[(i * (1, 0)) + (j * (0.5, ((-sqrt(3))) / 2))]], i, (-size), size]" a1="j" a2="(-size)" a3="size"/>
<output a0="list2"/>
</command>
<element type="list" label="list2">
<show object="true" label="false" ev="4"/>
<objColor r="128" g="0" b="128" alpha="0.1"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<expression label="text1" exp=""試著移動滑桿使得""/>
<element type="text" label="text1">
<show object="true" label="false" ev="43"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<startPoint x="-2.34" y="7.360000000000003" z="1"/>
</element>
<expression label="text6" exp=""$\overset{\large\rightharpoonup}{\scriptsize OQ} $$=x \overset{\large\rightharpoonup}{\small u}+y\overset{\large\rightharpoonup}{\small v} $""/>
<element type="text" label="text6">
<show object="true" label="false" ev="43"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<startPoint x="-2.32" y="7.020000000000003" z="1"/>
</element>
</construction>
</geogebra>