Geometric Demonstration of Cauchy-Schwarz Inequality Using Vectors (2)
2016-11-10 08:56
This GeoGebra example visually demonstrates the Cauchy-Schwarz inequality through vector geometry. It constructs geometric figures involving two vectors u and v along with their projections, utilizing orthogonal vectors, angle measurements, and segment length relationships to verify the core inequality |u·v| ≤ |u||v|. The construction incorporates vector operations, right triangle constructions, a
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<command name="GCD">
<input a0="v1" a1="v2"/>
<output a0="cd"/>
</command>
<element type="numeric" label="cd">
<value val="1"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="v11" exp="v1 / cd" />
<element type="numeric" label="v11">
<value val="17"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="v12" exp="v2 / cd" />
<element type="numeric" label="v12">
<value val="2"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="vnorm2" exp="sqrt(v11 / v12)" />
<element type="numeric" label="vnorm2">
<value val="2.9154759474226504"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="min1" exp="((a * b) * r^(2))" />
<element type="numeric" label="min1">
<value val="98"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="min2" exp="v1" />
<element type="numeric" label="min2">
<value val="17"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<command name="GCD">
<input a0="min1" a1="min2"/>
<output a0="mincd"/>
</command>
<element type="numeric" label="mincd">
<value val="1"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="min11" exp="min1 / mincd" />
<element type="numeric" label="min11">
<value val="98"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="min12" exp="min2 / mincd" />
<element type="numeric" label="min12">
<value val="17"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="text4" exp=""\left| \fgcolor{ff0000}{\vec{u}} \right|^2 = \left( \frac{ \fgcolor{660099}{ " + (LaTeX[r]) + " }}{ \left| \fgcolor{0000ff}{ \vec{v} } \right| \cos\theta } \right)^2 ≥ \frac{ \fgcolor{660099}{ " + (LaTeX[r]) + " ^2 }}{ \left| \fgcolor{0000ff}{ \vec{v} } \right|^2 } =" + (LaTeX[If[IsInteger[min11 / min12], min11 / min12 + "", "\frac{" + min11 + "}{" + min12 + "}"]]) + """ />
<element type="text" label="text4">
<show object="true" label="false" ev="8"/>
<condition showObject="showCoords"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="1"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="7" style="1"/>
<absoluteScreenLocation x="500" y="102"/>
</element>
<expression label="text5" exp=""\left| \fgcolor{0000ff}{\vec{v}} \right| = " + (LaTeX[If[IsInteger[vnorm2], vnorm2 + "", If[v12 ≟ 1, If[IsInteger[sqrt(v11)], sqrt(v11) + "", "\sqrt{" + v11 + "}"], "\frac{" + (If[IsInteger[sqrt(v11)], sqrt(v11) + "", "\sqrt{" + v11 + "}"]) + "}{" + (If[IsInteger[sqrt(v12)], sqrt(v12) + "", "\sqrt{" + v12 + "}"]) + "}"]]]) + """ />
<element type="text" label="text5">
<show object="true" label="false" ev="8"/>
<condition showObject="showCoords"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="2"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="7" style="1"/>
<absoluteScreenLocation x="320" y="103"/>
</element>
</construction>
</geogebra>