平面外积、有向面积与二阶行列式的几何演示
2016-12-02 01:16
本GeoGebra范例直观演示了平面向量的外积、有向面积与二阶行列式之间的几何联系。通过构造向量及其构成的多边形，动态展示向量夹角以及由两向量张成的平行四边形的有向面积。结合圆弧和角度标记，深入解析二阶行列式的几何意义及其正负号与向量旋转方向的关系，是理解线性代数与解析几何中向量运算的可视化工具。
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<startPoint exp="Point[c, 0.5]"/>
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<expression label="txH" exp=""高""/>
<element type="text" label="txH">
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<command name="ArcCounterClockwise">
<input a0="O" a1="O + n" a2="60"/>
<output a0="e" a1="b"/>
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<element type="conicpart" label="e">
<show object="true" label="false" ev="4"/>
<condition showObject="showN ∧ t ≟ 1"/>
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<labelMode val="0"/>
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<lineStyle thickness="2" type="20" typeHidden="1"/>
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<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
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<element type="vector" label="b">
<show object="true" label="false" ev="4"/>
<condition showObject="showN ∧ t ≟ 1"/>
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<command name="ArcCounterClockwise">
<input a0="O" a1="E" a2="60"/>
<output a0="h" a1="i"/>
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<element type="conicpart" label="h">
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<condition showObject="showN ∧ t ≟ 1"/>
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<lineStyle thickness="2" type="20" typeHidden="1"/>
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<element type="vector" label="i">
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<condition showObject="showN ∧ t ≟ 1"/>
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<expression label="txLen" exp=""長""/>
<element type="text" label="txLen">
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<condition showObject="showN ∧ t ≟ 1"/>
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<font serif="false" sizeM="1.4" size="7" style="1"/>
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<expression label="txPrj" exp=""投影量""/>
<element type="text" label="txPrj">
<show object="true" label="false" ev="40"/>
<condition showObject="showN ∧ t ≟ 1"/>
<objColor r="0" g="0" b="0" alpha="0"/>
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<font serif="false" sizeM="1.4" size="7" style="1"/>
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<expression label="text4" exp=""底 × 高""/>
<element type="text" label="text4">
<show object="true" label="false" ev="40"/>
<condition showObject="showN ∧ t ≟ 1"/>
<objColor r="0" g="0" b="0" alpha="0"/>
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<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="false" sizeM="1.4" size="7" style="1"/>
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<expression label="text5" exp=""= 長 × 投影量 \\ =\fgcolor{0000ff}{ \vec{v}} \cdot \fgcolor{008800}{ \vec{n}} \\ = " + (LaTeX[(n * v)]) + " \\ = \fgcolor{ff0000}{ \vec{u}} × \fgcolor{0000ff}{ \vec{v} }\\ = \begin{vmatrix} \fgcolor{ff0000}{ " + (LaTeX[x(u)]) + "}&\fgcolor{ff0000}{ " + (LaTeX[y(u)]) + "}\\ \fgcolor{0000ff}{ " + (LaTeX[x(v)]) + "}&\fgcolor{0000ff}{ " + (LaTeX[y(v)]) + "} \end{vmatrix}"" />
<element type="text" label="text5">
<show object="true" label="false" ev="8"/>
<condition showObject="showN ∧ t ≟ 1"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
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<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="7" style="1"/>
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</element>
</construction>
</geogebra>