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We are planning to add new Sec on Linear Algebra. #135

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We are planning to add new Sec on Linear Algebra.
mlamich1 Jul 26, 2024
00868cb
ON1_DotProductandAnglebetweentwovector (#149)
mlamich1 Jul 26, 2024
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42 changes: 40 additions & 2 deletions linear-algebra/source/future-ON/01.ptx
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Expand Up @@ -148,8 +148,32 @@ The dot product combines two vectors and creates a scalar that gives us geometri

<p>Need to create a segue to the fact that you can find the angle between vectors using the Law of Cosines as a starting place.</p>



<definition>
<statement>
<p> Given two vectors <m>\vec u</m> and <m>\vec v</m> in <m>\IR^n</m>, such that <m>\vec u</m> and <m>\vec v</m> are not parallel, let <m>\theta</m> be the <term>angle</term> between the two vectors,
then
<me>
\cos \theta = \frac{\vec u \cdot \vec v }{|\vec u||\vec v|}.
</me>
By the Law of Cosine:
<me>
|\vec u- \vec v|^2 = |\vec u|^2+|\vec v|^2- 2 |\vec u||\vec v| \cos \theta.
</me>
Using dot product of vector <m>\vec u-\vec v</m> with itself:

<md>
<mrow> |\vec u- \vec v|^2 \amp = (\vec u- \vec v) \cdot (\vec u- \vec v) \amp </mrow>
<mrow> \amp = \vec u \cdot \vec u - 2 ( \vec u \cdot \vec v) + \vec v \cdot \vec v \amp </mrow>
<mrow> \amp = |\vec u|^2 - 2 ( \vec u.\vec v) + |\vec v|^2 \amp </mrow>
</md>
Hence from above, we have:
<me>
\vec u \cdot \vec v = |\vec u||\vec v| \cos \theta .
</me>
</p></statement>
</definition>

<p> One of these activity should be deleted from here. </p>

<activity estimated-time="10">
<introduction>
Expand All @@ -159,8 +183,22 @@ The dot product combines two vectors and creates a scalar that gives us geometri
<task><p>Describe all vectors <m>\vec w</m> that are orthogonal to <m>\vec u</m>.</p></task>
<task><p>Find the angle between <m>\vec u</m> and <m>\vec v</m>.</p></task>
</activity>


<activity estimated-time="11">
<introduction>
<p>Consider two vectors <m>\vec u =\left[\begin{array}{c} 1 \\ 3 \\ 4 \\ -4 \end{array}\right]</m> and <m>\vec v = \left[\begin{array}{c} -1 \\ -3 \\ -4 \\ 4 \end{array}\right]</m>.</p>
</introduction>
<task><p>Use dot product to determine <m>|\vec u| </m> and <m> |\vec v|</m>.</p></task>
<task><p> Using dot product, find the distance between <m>\vec u</m> and <m>\vec v </m>.</p></task>
<task><p>Find the angle between <m>\vec u</m> and <m>\vec v</m>.</p></task>
</activity>

</subsection>
<subsection>



<title>Videos</title>
<!-- <figure>
<video xml:id="video-E3" href="https://southalabama.hosted.panopto.com/Panopto/Podcast/Social/1792c805-8df0-4b5c-a624-ac1901421d9d.mp4?mediaTargetType=videoPodcast">
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