Vector Decomposition and Solving Simultaneous Equations
2016-11-03 05:33
This example demonstrates how to decompose a vector into a linear combination of two basis vectors and solve for the decomposition coefficients using simultaneous equations. Using GeoGebra's vector, line, and intersection tools, it visually illustrates the geometric meaning and algebraic computation of vector decomposition, helping learners understand the connection between basis representation in
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</element>
<command name="Vector">
<input a0="A" a1="H"/>
<output a0="vAH"/>
</command>
<element type="vector" label="vAH">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="2.3" y="3.0900000000000003" z="0.0"/>
<lineStyle thickness="6" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
<startPoint exp="A"/>
</element>
<command name="Vector">
<input a0="I"/>
<output a0="w"/>
</command>
<element type="vector" label="w">
<show object="true" label="false" ev="11"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="0.5" y="1.5" z="0.0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
</element>
<expression label="text3" exp=""x""/>
<element type="text" label="text3">
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<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="1"/>
<startPoint exp="A + (x(I) * u)"/>
<labelOffset x="-7" y="5"/>
</element>
<expression label="text4" exp=""y""/>
<element type="text" label="text4">
<show object="true" label="false" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="1"/>
<startPoint exp="A + (y(I) * v)"/>
<labelOffset x="-28" y="-11"/>
</element>
</construction>
</geogebra>