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9.3 - Projections
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<h2>[9.3] - Projections &amp; dot product
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<p class="show"><b>Read</b> section 9.3 in the textbook and answer these questions.
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<li>For the two vectors shown, draw the vector projection of $\myv b$ onto $\myv a$.<br><img src="0g/9.3.ab.jpg">
<li>For the same two vectors, draw the vector projection of $\myv a$ onto $\myv b$.<br><img src="0g/9.3.ab.jpg">
<li>What is the angle between the vectors $\langle 1,0,-1\rangle$ and 
$\langle 1,1,0\rangle$?
<p class="justme result">$$\theta = \arccos(\myv a\cdot\myv b/(ab))=\arccos\left(\frac 1{\sqrt 2*\sqrt 2}\right)=\pi/3= 60{}^o$$
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<li>Suppose that $\myv a \neq 0$ and 
$\myv a \cdot \myv b=\myv a \cdot \myv c$.
  This means that $\myv b$ and $\myv c$ have the same projection onto $\myv a$.  Does this automatically mean that $\myv b = \myv c$?<br><br>

Consider $\myv a=\langle 1,2 \rangle$ and $\myv b=\langle 2,1 \rangle$.  Show that the answer to the question above is "no", by finding a vector $\myv c$ which is not equal to $\myv b$, but never the less satisfies $\myv a \cdot \myv b=\myv a \cdot \myv c$.


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