Geometric Demonstration of the Cauchy-Schwarz Inequality (4)
2016-11-11 02:19
This GeoGebra applet demonstrates the Cauchy-Schwarz Inequality using vector geometry. By constructing vectors and angles, it illustrates that the absolute value of the dot product of two vectors is bounded by the product of their magnitudes. Users can interactively observe how the angle affects the inequality, providing a clear visualization of this algebraic concept.
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<command name="Segment">
<input a0="O + (offSeg * un)" a1="A + (offSeg * un)"/>
<output a0="segOA"/>
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<element type="segment" label="segOA">
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<expression label="uv" exp="sqrt(((u * u) * (v * v)))" />
<element type="numeric" label="uv">
<value val="31.41570047925743"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.1"/>
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<expression label="text3" exp=""\left| \fgcolor{ff0000}{\vec{u}} \right| \left| \fgcolor{0000ff}{\vec{v}} \right| = \frac{\fgcolor{ff0066}{" + (LaTeX[(v * u)]) + " }}{\cos\theta}= " + (LaTeX[uv]) + """ />
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<condition showObject="showCoords"/>
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<expression label="text1" exp=""(\fgcolor{ff0000}{" + (LaTeX[If[m ≟ 1, "", m^(2) + ""]]) + "}a+\fgcolor{ee6600}{" + (LaTeX[If[n ≟ 1, "", n^(2) + ""]]) + "}b) \left(\frac{\fgcolor{006600}{" + (LaTeX[p^(2)]) + " }}{a}+\frac{\fgcolor{0000ff}{" + (LaTeX[q^(2)]) + " }}{b}\right) =\fgcolor{6600ff}{" + (LaTeX[uv^(2)]) + "}"" />
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<expression label="text4" exp=""註：座標沒有正常切換時 請拉動下面的滑桿。""/>
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</construction>
</geogebra>