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EV5: add doenet basis visualization #881

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fabb36c
add doenet basis visualization
StevenClontz Oct 3, 2025
f64de61
tweaks
StevenClontz Oct 3, 2025
d078ede
tweak interactive
StevenClontz Oct 3, 2025
768a4e7
and another tweak
StevenClontz Oct 3, 2025
59b5c60
update doenet interactive
StevenClontz Oct 3, 2025
8872c7d
separate tool and activity
StevenClontz Oct 3, 2025
174f964
fix xref
StevenClontz Oct 3, 2025
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31 changes: 31 additions & 0 deletions source/linear-algebra/source/02-EV/05.ptx
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Expand Up @@ -489,6 +489,37 @@ Not a basis, because not only does it fail to span <m>\IR^4</m>, it's also linea
</statement>
</fact>

<remark xml:id="EV5-basis-visualization-tool">
<p>
A basis for a Euclidean vector space can be used to create alternative coordinate systems, as can
be visualized by the following tool.
Comment on lines +494 to +495
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@jkostiuk jkostiuk Dec 15, 2025

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Suggested change
A basis for a Euclidean vector space can be used to create alternative coordinate systems, as can
be visualized by the following tool.
A basis for a Euclidean vector space can be used to create alternative coordinate systems. This is visualized in the following interactive. As you drag the vector <m>\vec{v}</m> around the plane, the interactive shows how <m>\vec{v}</m> is expressed as a linear combination of the basis vectors <m>\vec{e}_1</m> and <m>\vec{e}_2</m>. If you click the toggle for the custom basis <m>\{\vec{b}_1,\vec{b}_2\}, the interactive will now show how to express the vector <m>\vec{v}</m> as a linear combination of the vectors <m>\vec{b}_1</m> and <m>\vec{b}_2</m>. You may also drag the vectors <m>\vec{b}_1</m> and <m>\vec{b}_2</m> to see how the results vary with different choices of custom bases.

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@jkostiuk jkostiuk Dec 15, 2025

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@StevenClontz I like the interactive a lot. I think a caption that describes what the interactive does would be helpful. I've provided a suggestion for your review.

</p>
<interactive label="EV5-basis-visualization" platform="doenetml" width="90%" aspect="3:4">
<slate surface="doenetml">
<xi:include parse="text" href="doenet/EV5-basis-visualization.xml"/>
</slate>
</interactive>
</remark>

<activity>
<statement>
<p>
Using the tool from <xref ref="EV5-basis-visualization-tool"/>,
find the unique way to write the vector
<m>\vec v = \left[\begin{array}{c}-7\\3\end{array}\right]</m>
as a linear combination of vectors from the basis <m>\{\vec b_1,\vec b_2\}</m> where
<m>\vec b_1=\left[\begin{array}{c}-2\\2\end{array}\right]</m> and
<m>\vec b_2=\left[\begin{array}{c}3\\1\end{array}\right]</m>.
<ol marker="A." cols="2">
<li><m>\vec v = 2\vec b_1-\vec b_2</m></li>
<li><m>\vec v = -2\vec b_1-2\vec b_2</m></li>
<li><m>\vec v = -\vec b_1+2\vec b_2</m></li>
<li><m>\vec v = \vec b_1-\vec b_2</m></li>
</ol>
</p>
</statement>
</activity>

</subsection>

<subsection>
Expand Down
65 changes: 65 additions & 0 deletions source/linear-algebra/source/02-EV/doenet/EV5-basis-visualization.xml
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@@ -0,0 +1,65 @@
<!-- https://doenet.org/portfolioeditor/_PSDg7v5ugo6RBMxFARE3Z/_8sE7bqhRzNAnb33Xfp54o -->

<setup>
<boolean name="standardBasis">$basisChoice.selectedIndices = 1</boolean>
<point name="v">(-2,-3)
<constraints>
<constrainToGrid/>
</constraints>
</point>
<point name="b2">(-1,3)
<constraints>
<constrainToGrid/>
</constraints>
</point>
<point name="b1">(4,1)
<constraints>
<constrainToGrid/>
</constraints>
</point>
<math name="x" simplify>($v.x*$b2.y-$v.y*$b2.x)/($b1.x*$b2.y-$b2.x*$b1.y)</math>
<math name="y" simplify>(-$v.x*$b1.y+$v.y*$b1.x)/($b1.x*$b2.y-$b2.x*$b1.y)</math>
</setup>

<p>
<choiceInput name="basisChoice" preselectChoice="1">
<choice>Standard basis {e₁,e₂}</choice>
<choice>Custom basis {b₁,b₂}</choice>
</choiceInput>
</p>

<graph xmin="-8" ymin="-8" xmax="8" ymax="8" displayXAxis="$standardBasis" displayYAxis="$standardBasis">
<!-- both views -->
<point coords="$v" styleNumber="1">
<label><m>\vec v</m></label>
</point>
<lineSegment draggable="false" endpoints="(0,0) $v" styleNumber="1"/>
<point coords="$b1" styleNumber="2">
<label><m>\vec b_1</m></label>
</point>
<lineSegment draggable="false" endpoints="(0,0) $b1" styleNumber="2"/>
<point coords="$b2" styleNumber="3">
<label><m>\vec b_2</m></label>
</point>
<lineSegment draggable="false" endpoints="(0,0) $b2" styleNumber="3"/>
<!-- standard basis -->
<point draggable="false" hide="not $standardBasis" coords="($v.x,0)" styleNumber="1">
<label><m>$v.x\vec e_1</m></label>
</point>
<lineSegment draggable="false" hide="not $standardBasis" endpoints="($v.x,0) $v" styleNumber="6"/>
<point draggable="false" hide="not $standardBasis" coords="(0,$v.y)" styleNumber="1">
<label><m>$v.y\vec e_2</m></label>
</point>
<lineSegment draggable="false" hide="not $standardBasis" endpoints="($0,$v.y) $v" styleNumber="6"/>
<!-- nonstandard basis -->
<line draggable="false" through="(0,0) $b1" styleNumber="2" hide="$standardBasis"/>
<line draggable="false" through="(0,0) $b2" styleNumber="3" hide="$standardBasis"/>
<point draggable="false" hide="$standardBasis" coords="$x*$b1" styleNumber="1">
<label><m>$x\vec b_1</m></label>
</point>
<lineSegment draggable="false" hide="$standardBasis" endpoints="$x*$b1 $v" styleNumber="6"/>
<point draggable="false" hide="$standardBasis" coords="$y*$b2" styleNumber="1">
<label><m>$y\vec b_2</m></label>
</point>
<lineSegment draggable="false" hide="$standardBasis" endpoints="$y*$b2 $v" styleNumber="6"/>
</graph>