Derivatives and Integrals: An Annotated Discourse

<worksheet label="worksheet-dot-products" courseid="MAT-150" series="Activity" seriescode="13">
  <title>Dot products and projection</title>
  <page>
  <sidebyside width="45%" margins="0%">
    <exercise>
      <introduction>
        <p>
          Let <m>{\vec v}_1 = (-4,1)</m>,
          <m>{\vec v}_2 = (2,2)</m>,
          <m>{\vec v}_3 = (1,2,3)</m>, <m>{\vec v}_4 = (-2,1,0)</m>.
          Find the values of the following expressions:
        </p>
      </introduction>
      <task workspace="1in">
        <p>
          <m>{\vec v}_1 \cdot {\vec v}_2 = <fillin /></m>
        </p>
      </task>
      <task workspace="1.0in">
        <p>
          <m>{\vec v}_3 \cdot {\vec v}_4 = <fillin /></m>
        </p>
      </task>
      <task workspace="1in">
        <p>
          <m>\lVert{\vec v}_1\rVert = <fillin /></m>
        </p>
      </task>
      <task workspace="1in">
        <p>
          <m>\lVert{\vec v}_4\rVert = <fillin /></m>
        </p>
      </task>
      <task workspace="1in">
        <p>
          Are any of these vectors perpendicular to each other?
          <fillin />
        </p>
      </task>
    </exercise>
    <exercise workspace="3in">
      <statement>
        <p>
          The vectors <m>\vec a = (3,9)</m> and <m>\vec u = (4,2)</m> are pictured below.
          Derive the formula for projection on a line and use it to find the projection of <m>\vec a</m> on the line spanned by <m>\vec u</m>.
          Also compute the length of the residual vector.
        </p>
        <image width="100%" source="images/projection1.png">
          <shortdescription>two vectors in a Cartesian plane</shortdescription>
        </image>
      </statement>
    </exercise>
  </sidebyside>
  </page>
  <page>
  <sidebyside width="48%" margins="0%">
    <exercise workspace="1.25in">
      <introduction>
        <p>
          Consider the vector equation
          <me>
            m \begin{bmatrix}2 \\ 5\end{bmatrix} = \begin{bmatrix}3 \\ 7\end{bmatrix}
          </me>.
        </p>
      </introduction>
      <task>
        <p>
          Check that there is no solution <m>m</m> that makes the equation true.
        </p>
      </task>
      <task>
        <p>
          Use projection to find the best approximation <m>\hat m</m>.
        </p>
      </task>
      <task>
        <p>
          Compute <m>\hat m \begin{bmatrix}2 \\ 5\end{bmatrix} </m>.
        </p>
      </task>
      <task>
        <p>
          Compute the residual vector.
        </p>
      </task>
      <task>
        <p>
          Compute the length of the residual vector and explain what it means.
        </p>
      </task>
    </exercise>
    <exercise>
      <introduction>
        <p>
          Consider the system of equations
          <md>
            <mrow>3t \amp =5</mrow>
            <mrow>2t \amp = 9</mrow>
          </md>.
        </p>
      </introduction>
      <task workspace="2in">
        <p>
          Write the system in vector form.
        </p>
      </task>
      <task workspace="3.9in">
        <p>
          Find the best estimate, <m>\hat t</m>,
          of <m>t</m> using projection.
        </p>
      </task>
      <task workspace="2in">
        <p>
          Compute the length of the residual vector.
        </p>
      </task>
    </exercise>
  </sidebyside>
  </page>
</worksheet>