向量的垂直分解
2013-11-13 15:14
本GeoGebra范例演示了向量的垂直分解（正交分解）过程。通过构建平行四边形和直角三角形，展示如何将一个给定向量分解为两个相互垂直的分向量。界面包含向量、线段及LaTeX分数公式，直观呈现向量在特定方向及其法向量方向上的投影与分量计算，适用于线性代数与解析几何中向量运算的教学与学习。
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.290.0" id="c0396c6b-e424-44c3-9caf-2555629168a6" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="950" height="650" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.9978947368421053" orientation="1" />
</panes>
<views>
<view id="4" toolbar="0 59 || 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="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 | 1011 || 1008 1009 || 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="true" location="3" size="948" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="false" location="3" size="226" window="100,100,250,400" />
<view id="16" visible="false" inframe="false" stylebar="false" location="1" size="150" 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="4097" visible="false" inframe="true" stylebar="true" location="1" size="390" window="503,130,700,543" />
<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="false" stylebar="false" location="1" size="300" window="50,50,500,500" />
</views>
<toolbar show="false" items="0 39 59 | 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 66 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="3"/>
<font size="12"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="948" height="648"/>
<coordSystem xZero="439.99999999999966" yZero="453.75" scale="49.99999999999998" yscale="50.00000000000002"/>
<evSettings axes="false" grid="true" gridIsBold="false" pointCapturing="2" rightAngleStyle="1" checkboxSize="26" gridType="0"/>
<bgColor r="255" g="255" b="255"/>
<axesColor r="0" g="0" b="0"/>
<gridColor r="0" g="0" b="0"/>
<lineStyle axes="1" grid="20"/>
<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="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="0.1" speed="1" type="1" playing="false"/>
<coords x="5" y="6" z="1"/>
<pointSize val="5"/>
<pointStyle val="2"/>
</element>
<element type="point" label="A">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="3" y="1" z="1"/>
<pointSize val="5"/>
<pointStyle val="2"/>
</element>
<command name="Vector">
<input a0="A"/>
<output a0="a"/>
</command>
<element type="vector" label="a">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="3" y="1" z="0"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
</element>
<expression label="n" exp="((-y(a)), x(a))" type="vector" />
<element type="vector" label="n">
<show object="true" label="false" ev="4"/>
<objColor r="51" g="153" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="-1" y="3" z="0"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
</element>
<expression label="u1" exp="a / Length[a]" type="vector" />
<element type="vector" label="u1">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="0.9486832980505138" y="0.31622776601683794" z="0"/>
<lineStyle thickness="3" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
</element>
<expression label="v1" exp="n / Length[n]" type="vector" />
<element type="vector" label="v1">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="-0.31622776601683794" y="0.9486832980505138" z="0"/>
<lineStyle thickness="3" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
</element>
<command name="Vector">
<input a0="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"/>
<selectionAllowed val="false"/>
<coords x="5" y="6" z="0"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
</element>
<expression label="s" exp="(v * u1)" />
<element type="numeric" label="s">
<value val="6.640783086353597"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<selectionAllowed val="false"/>
</element>
<expression label="t" exp="(v * v1)" />
<element type="numeric" label="t">
<value val="4.110960958218892"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<selectionAllowed val="false"/>
</element>
<element type="point" label="O">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<selectionAllowed val="false"/>
<coords x="0" y="0" z="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<expression label="au" exp="(s * u1)" type="vector" />
<element type="vector" label="au">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="102" b="153" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="6.3" y="2.1" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
</element>
<expression label="av" exp="(t * v1)" type="vector" />
<element type="vector" label="av">
<show object="true" label="false" ev="4"/>
<objColor r="102" g="204" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="-1.2999999999999998" y="3.899999999999999" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
</element>
<command name="Segment">
<input a0="O + v" a1="O + au"/>
<output a0="b"/>
</command>
<element type="segment" label="b">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="3.9" y="1.2999999999999998" z="-27.299999999999997"/>
<lineStyle thickness="2" type="10" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="O + v" a1="O + av"/>
<output a0="c"/>
</command>
<element type="segment" label="c">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="2.100000000000001" y="-6.3" z="27.299999999999997"/>
<lineStyle thickness="2" type="10" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="txtU" exp=""\vec{a}""/>
<element type="text" label="txtU">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="2" size="12" style="1"/>
<startPoint exp="O + a"/>
<labelOffset x="11" y="-25"/>
</element>
<expression label="txtV" exp=""\vec{b}""/>
<element type="text" label="txtV">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="2" size="12" style="1"/>
<startPoint exp="O + v"/>
<labelOffset x="-6" y="-44"/>
</element>
<expression label="txtW" exp=""\vec{n}""/>
<element type="text" label="txtW">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="2" size="12" style="1"/>
<startPoint exp="O + n"/>
<labelOffset x="-24" y="-35"/>
</element>
<element type="image" label="pic1">
<file name="65fa5a523f4b4eb7a49a1d5451c69200/bullet_star.png"/>
<inBackground val="false"/>
<absoluteScreenLocation x="13" y="274"/> <show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="1"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<selectionAllowed val="false"/>
</element>
<expression label="x1" exp="((v * a)) / ((a * a))" />
<element type="numeric" label="x1">
<value val="2.1"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.1"/>
<selectionAllowed val="false"/>
</element>
<expression label="y1" exp="((v * n)) / ((a * a))" />
<element type="numeric" label="y1">
<value val="1.3"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.1"/>
<selectionAllowed val="false"/>
</element>
<expression label="text8" exp=""\vec{n}=" + (LaTeX[(x(n), y(n))]) + """ />
<element type="text" label="text8">
<show object="true" label="false" ev="8"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="2" size="12" style="1"/>
<absoluteScreenLocation x="17" y="122"/>
</element>
<expression label="text1" exp=""\vec{a}=" + (LaTeX[LaTeX[(x(a), y(a))]]) + """ />
<element type="text" label="text1">
<show object="true" label="false" ev="8"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="2" size="12" style="1"/>
<absoluteScreenLocation x="18" y="24"/>
</element>
<command name="LatexFraction">
<input a0="(v * a)" a1="(a * a)" a2="""" a3="""" a4="false"/>
<output a0="latexX"/>
</command>
<element type="text" label="latexX">
<symbolic val="true" />
<show object="false" label="true" ev="8"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="false" sizeM="1" size="0" style="0"/>
<decimals val="2"/>
<startPoint x="0" y="0" z="1"/>
</element>
<command name="LatexFraction">
<input a0="(v * n)" a1="(a * a)" a2="""" a3="""" a4="false"/>
<output a0="latexY"/>
</command>
<element type="text" label="latexY">
<symbolic val="true" />
<show object="false" label="true" ev="8"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="false" sizeM="1" size="0" style="0"/>
<decimals val="2"/>
<startPoint x="0" y="0" z="1"/>
</element>
<expression label="a0a" exp="(a * a)" />
<element type="numeric" label="a0a">
<value val="10"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.1"/>
<selectionAllowed val="false"/>
</element>
<expression label="v0a" exp="(v * a)" />
<element type="numeric" label="v0a">
<value val="21"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.1"/>
<selectionAllowed val="false"/>
</element>
<expression label="v0n" exp="(v * n)" />
<element type="numeric" label="v0n">
<value val="13"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.1"/>
<selectionAllowed val="false"/>
</element>
<expression label="txtSU" exp=""" + (LaTeX[LatexFractionCoefficient[(v * a), (a * a), "", "", "\vec{a}", false, false]]) + " "" />
<element type="text" label="txtSU">
<show object="true" label="false" ev="4"/>
<condition showObject="d"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="1"/>
<startPoint exp="O + au"/>
<labelOffset x="-5" y="1"/>
</element>
<expression label="txtTV" exp=""" + (LaTeX[LatexFractionCoefficient[(v * n), (a * a), "", "", "\vec{n}", false, false]]) + " "" />
<element type="text" label="txtTV">
<show object="true" label="false" ev="4"/>
<condition showObject="d"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="1"/>
<startPoint exp="O + av"/>
<labelOffset x="-61" y="-11"/>
</element>
<expression label="red" exp=""ff0000""/>
<element type="text" label="red">
<symbolic val="true" />
<show object="false" label="false" ev="8"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<startPoint x="0.24000000000000243" y="4.084999999999999" z="1"/>
</element>
<expression label="blue" exp=""0000ff""/>
<element type="text" label="blue">
<symbolic val="true" />
<show object="false" label="false" ev="8"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<startPoint x="0.24000000000000243" y="4.084999999999999" z="1"/>
</element>
<expression label="text2" exp=""\vec{b}=" + (LaTeX[LaTeX[(x(v), y(v))]]) + """ />
<element type="text" label="text2">
<show object="true" label="false" ev="8"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="2" size="12" style="1"/>
<absoluteScreenLocation x="18" y="73"/>
</element>
<expression label="text3" exp=""\fgcolor{ff0000}{\vec{a}} \times \fgcolor{0000ff}{\vec{b}}= \left\vert \begin{array}{} " + (LaTeX[x(a)]) + " & " + (LaTeX[y(a)]) + " \\ " + (LaTeX[x(v)]) + " & " + (LaTeX[y(v)]) + " \\ \end{array} \right\vert = " + (LaTeX[(v * n)]) + """ />
<element type="text" label="text3">
<show object="true" label="false" ev="8"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="1"/>
<absoluteScreenLocation x="20" y="273"/>
</element>
<expression label="txB2" exp=""\fgcolor{0000ff}{\vec{b}}= " + (LaTeX[LatexFractionCoefficient[(v * a), (a * a), "", "", "\fgcolor{ff0000}{\vec{a}}", false, false]]) + " " + (LaTeX[LatexFractionCoefficient[(v * n), (a * a), "", "", "\fgcolor{006600}{\vec{n}}", true, false]]) + """ />
<element type="text" label="txB2">
<show object="true" label="false" ev="8"/>
<condition showObject="d"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="2" size="12" style="1"/>
<absoluteScreenLocation x="18" y="174"/>
</element>
<expression label="text9" exp=""\fgcolor{0000ff}{\vec{b}}= \left ( \frac{\vec{a} \cdot \fgcolor{" + (LaTeX[blue]) + "}{\vec{b}} }{\vec{a}\cdot\vec{a}} \right ) \fgcolor{ff0000}{\vec{a}} + \left ( \frac{ \vec{a} \times \fgcolor{0000ff}{\vec{b}} }{\vec{a}\cdot\vec{a}} \right ) \fgcolor{006600}{\vec{n}}"" />
<element type="text" label="text9">
<show object="true" label="false" ev="8"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="2" size="12" style="1"/>
<absoluteScreenLocation x="269" y="24"/>
</element>
<element type="boolean" label="d">
<value val="true"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="735" y="49"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<caption val="顯示分量"/>
</element>
</construction>
</geogebra>