Division Point and Vector Ratio
2021-07-26 03:09
This GeoGebra example demonstrates the concept of division point and vector ratio in geometry. By drawing line segments, points, and vectors, and calculating vector ratios, the application of vectors in geometry can be intuitively understood. The example includes the calculation of the midpoint of a line segment, unit vectors, and the application of vector ratios, suitable for teaching and self-st
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.654.0" app="classic" platform="w" id="d16cbd5c-3384-450e-93c2-6ebf720bae97" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/apps/xsd/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="808" height="391" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="1" orientation="1" />
</panes>
<views>
<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="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="true" location="3" size="808" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="false" location="3" size="200" tab="ALGEBRA" window="100,100,600,400" />
<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="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="128" visible="false" inframe="false" stylebar="false" location="1" size="480" window="50,50,500,500" />
<view id="4097" visible="false" inframe="false" stylebar="true" location="1" size="462" window="580,250,700,550" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="false" stylebar="true" location="1" size="150" 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 | 100002 100001" position="1" help="false" />
<input show="false" cmd="true" top="algebra" />
<dockBar show="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="16"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="807" height="390"/>
<coordSystem xZero="263.9999999999999" yZero="164.50000000000009" scale="50.00000000000002" yscale="49.99999999999991"/>
<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>
<tableview min="-2" max="2" step="1"/>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<element type="point" label="A">
<show object="true" label="true" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="-10" y="-4"/>
<labelMode val="0"/>
<fixed val="true"/>
<selectionAllowed val="false"/>
<coords x="0" y="0" z="1"/>
<pointSize val="6"/>
<pointStyle val="0"/>
</element>
<element type="point" label="B">
<show object="true" label="true" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="-8" y="-4"/>
<labelMode val="0"/>
<fixed val="true"/>
<selectionAllowed val="false"/>
<coords x="4" y="0" z="1"/>
<pointSize val="6"/>
<pointStyle val="0"/>
</element>
<command name="Line">
<input a0="A" a1="B"/>
<output a0="lineAB"/>
</command>
<element type="line" label="lineAB">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="153" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="0" y="4" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="Point">
<input a0="lineAB"/>
<output a0="X"/>
</command>
<element type="point" label="X">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="-9" y="-4"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="5.94" y="0" z="1"/>
<pointSize val="6"/>
<pointStyle val="0"/>
</element>
<command name="VectorRatio">
<input a0="Vector[A, X]" a1="Vector[X, B]"/>
<output a0="r"/>
</command>
<element type="numeric" label="r">
<value val="-3.0618556701030926"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<selectionAllowed val="false"/>
</element>
<expression label="t" exp="2"/>
<element type="numeric" label="t">
<value val="2"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<animation speed="1" type="0" playing="false"/>
<selectionAllowed val="false"/>
</element>
<command name="UnitVector">
<input a0="Vector[OrthogonalVector[Vector[A, B]]]"/>
<output a0="n"/>
</command>
<element type="vector" label="n">
<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" y="1" z="0"/>
<lineStyle thickness="9" type="0" typeHidden="1" opacity="127"/>
<coordStyle style="cartesian"/>
<startPoint exp="A"/>
</element>
<expression label="pad" exp="0.2"/>
<element type="numeric" label="pad">
<value val="0.2"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<animation speed="1" type="0" playing="false"/>
<selectionAllowed val="false"/>
</element>
<expression label="v1" exp="(pad * 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" y="0.2" z="0"/>
<lineStyle thickness="9" type="0" typeHidden="1" opacity="127"/>
<coordStyle style="cartesian"/>
</element>
<expression label="v2" exp="((sgn(r) * pad) * n)" type="vector" />
<element type="vector" label="v2">
<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" y="-0.2" z="0"/>
<lineStyle thickness="9" type="0" typeHidden="1" opacity="127"/>
<coordStyle style="cartesian"/>
</element>
<command name="Vector">
<input a0="X + v2" a1="B + v2"/>
<output a0="XB"/>
</command>
<element type="vector" label="XB">
<show object="true" label="false" ev="4"/>
<objColor r="102" g="153" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="-1.9400000000000004" y="0" z="0"/>
<lineStyle thickness="9" type="0" typeHidden="1" opacity="127"/>
<coordStyle style="cartesian"/>
<startPoint x="5.94" y="-0.2" z="1"/>
</element>
<expression label="text1" exp=""" + (LaTeX[r]) + """ />
<element type="text" label="text1">
<show object="true" label="false" ev="20"/>
<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" size="0" style="1"/>
<decimals val="2"/>
<startPoint exp="X"/>
<labelOffset x="-9" y="-57"/>
</element>
<command name="Midpoint">
<input a0="A" a1="B"/>
<output a0="C"/>
</command>
<element type="point" label="C">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="2" y="0" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<expression label="text2" exp=""1""/>
<element type="text" label="text2">
<show object="true" label="false" ev="20"/>
<objColor r="128" g="128" b="128" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="1"/>
<startPoint exp="C"/>
<labelOffset x="-10" y="22"/>
</element>
<expression label="A1" exp="A" type="point" />
<element type="point" label="A1">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="204" b="51" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="0" y="0" z="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<expression label="B1" exp="B" type="point" />
<element type="point" label="B1">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="204" b="51" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="4" y="0" z="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<expression label="text3" exp=""0""/>
<element type="text" label="text3">
<show object="true" label="false" ev="20"/>
<objColor r="128" g="128" b="128" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="1"/>
<startPoint exp="A"/>
<labelOffset x="-11" y="21"/>
</element>
<expression label="eq1" exp="x = x(Corner[1])" type="line" />
<element type="line" label="eq1">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<selectionAllowed val="false"/>
<coords x="1" y="0" z="5.299999999999995"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="lineAB" a1="eq1"/>
<output a0="D"/>
</command>
<element type="point" label="D">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="-21.19999999999998" y="0" z="4"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<expression label="text4" exp=""←-1^{+}""/>
<element type="text" label="text4">
<show object="true" label="false" ev="20"/>
<objColor r="128" g="128" b="128" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="1"/>
<startPoint exp="D"/>
<labelOffset x="-3" y="18"/>
</element>
<expression label="text5" exp=""+∞""/>
<element type="text" label="text5">
<show object="true" label="false" ev="20"/>
<objColor r="128" g="128" b="128" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="1"/>
<startPoint exp="B"/>
<labelOffset x="-50" y="21"/>
</element>
<expression label="text6" exp=""-∞""/>
<element type="text" label="text6">
<show object="true" label="false" ev="20"/>
<objColor r="128" g="128" b="128" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="1"/>
<startPoint exp="B"/>
<labelOffset x="3" y="22"/>
</element>
<expression label="eq2" exp="x = x(Corner[2])" type="line" />
<element type="line" label="eq2">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<selectionAllowed val="false"/>
<coords x="1" y="0" z="-10.879999999999997"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="eq2" a1="lineAB"/>
<output a0="E"/>
</command>
<element type="point" label="E">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="43.51999999999999" y="0" z="4"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<expression label="text7" exp=""→-1^{-}""/>
<element type="text" label="text7">
<show object="true" label="false" ev="20"/>
<objColor r="128" g="128" b="128" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="1"/>
<startPoint exp="E"/>
<labelOffset x="-71" y="17"/>
</element>
<expression label="pad2" exp="1"/>
<element type="numeric" label="pad2">
<value val="1"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<animation speed="1" type="0" playing="false"/>
<selectionAllowed val="false"/>
</element>
<expression label="f" exp="f(x) = ((-x)) / (x - 4)" type="function"/>
<element type="function" label="f">
<show object="true" label="false" ev="4"/>
<condition showObject="showFunc"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<selectionAllowed val="false"/>
<lineStyle thickness="4" type="10" typeHidden="1" opacity="178"/>
</element>
<expression label="h" exp="y = -1" type="line"/>
<element type="line" label="h">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="153" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<selectionAllowed val="false"/>
<coords x="0" y="1" z="1"/>
<lineStyle thickness="2" type="10" typeHidden="1" opacity="178"/>
<eqnStyle style="explicit"/>
</element>
<command name="Line">
<input a0="B + (pad2 * n)" a1="B - (pad2 * n)"/>
<output a0="g"/>
</command>
<element type="line" label="g">
<show object="true" label="false" ev="4"/>
<objColor r="160" g="160" b="160" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="2" y="0" z="-8"/>
<lineStyle thickness="3" type="20" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="Vector">
<input a0="A + v1" a1="X + v1"/>
<output a0="XA"/>
</command>
<element type="vector" label="XA">
<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="5.94" y="0" z="0"/>
<lineStyle thickness="9" type="0" typeHidden="1" opacity="127"/>
<coordStyle style="cartesian"/>
<startPoint x="0" y="0.2" z="1"/>
</element>
<expression label="text8" exp=""\frac{\overrightarrow{A\color{red}{X}}}{\overrightarrow{{\color{red}X}B}}=" + (LaTeX[r]) + """ />
<element type="text" label="text8">
<show object="true" label="false" ev="8"/>
<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="1"/>
<startPoint x="-4.540000000000001" y="2.2999999999999954" z="1"/>
</element>
<element type="boolean" label="showFunc">
<value val="true"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="37" y="280"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<caption val="顯示函數圖形"/>
</element>
</construction>
</geogebra>