vectors.xml

<?xml version="1.0" encoding="UTF-8"?>

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<section xml:id="vectors">
  <title>Vectors</title>

  <objectives>
    <ol>
      <li>Learn how to add and scale vectors in <m>\R^n</m>, both algebraically and geometrically.</li>
      <li>Understand linear combinations geometrically.</li>
      <li><em>Pictures:</em> vector addition, vector subtraction, linear combinations.</li>
      <li><em>Vocabulary words:</em> <term>vector</term>, <term>linear combination</term>.</li>
    </ol>
  </objectives>

  <subsection>
    <title>Vectors in <m>\R^n</m></title>

    <p>We have been drawing points in <m>\R^n</m> as dots in the line, plane, space, etc.  We can also draw them as <em>arrows</em>.  Since we have two geometric interpretations in mind, we now discuss the relationship between the two points of view.</p>

    <definition hide-type="true">
      <title>Points and Vectors</title>
      <idx><h>Point</h></idx>
      <idx><h>Vector</h><h>definition of</h></idx>
      <statement>
        <p>Again, a <term>point</term> in <m>\R^n</m> is drawn as a dot.
        <latex-code>
              <![CDATA[
    \begin{tikzpicture}[scale=1, every pin/.style={whitebg, thin border}]
    \draw[grid lines] (-1,-1) grid (4, 4);
    \draw[->, thick] (-1,0) -- (4,0);
    \draw[->, thick] (0,-1) -- (0,4);
    \point[seq-blue, pin={
      [pin edge={seq-blue}, text=seq-blue]
      -85:the point $(1,3)$}] (x) at (1,3);
    \end{tikzpicture}
              ]]>
        </latex-code>
        A <term>vector</term> is a point in <m>\R^n</m>, drawn as an arrow.
        <latex-code>
              <![CDATA[
    \begin{tikzpicture}[scale=1, every pin/.style={whitebg, thin border}]
    \draw[grid lines] (-1,-1) grid (4, 4);
      \draw[thick vector, green!50!black] (0,0)
        -- node[midway, pin={[right]-10:the vector $1\choose3$}] {} (1,3);
  \end{tikzpicture}
              ]]>
        </latex-code>
        </p>
      </statement>
    </definition>

    <p>The difference is purely psychological: <em>points and vectors are both just lists of numbers</em>.</p>

    <example hide-type="true">
      <title>Interactive: A vector in <m>\R^3</m>, by coordinates</title>
      <figure>
        <caption>A vector in <m>\R^3</m>, and its coordinates.  Drag the arrow head and tail.</caption>
        <mathbox source="demos/vector.html" height="400px"/>
      </figure>
    </example>

    <p>When we think of a point in <m>\R^n</m> as a vector, we will usually write it vertically, like a matrix with one column:
    <me>v = \vec{1 3}.</me>
    We will also write <m>0</m> for the zero vector.
    <notation><usage>{1\choose 2}</usage><description>A vector</description></notation>
    <notation><usage>0</usage><description>The zero vector</description></notation>
    </p>

    <p>Why make the distinction between points and vectors?  A vector need not start at the origin: <em>it can be located anywhere</em>! In other words, an arrow is determined by its length and its direction, not by its location.  For instance, these arrows all represent the vector <m>\color{seq-green}\vec{1 2}</m>.
    <latex-code>
          <![CDATA[
\begin{tikzpicture}
  \draw[help lines] (-1,-1) grid (4, 4);
  \draw[thick vector, seq-green] (0,1) -- (1,3);
  \draw[thick vector, seq-green] (1,1) -- (2,3);
  \draw[thick vector, seq-green] (2,0) -- (3,2);
\end{tikzpicture}
          ]]>
    </latex-code></p>

    <bluebox>
      <p>Unless otherwise specified, we will assume that all vectors start at the origin.
      </p>
    </bluebox>

    <p>Vectors makes sense in the real world: many physical quantities, such as velocity, are represented as vectors.  But it makes more sense to think of the velocity of a car as being located at the car.</p>

    <remark>
      <p>Some authors use boldface letters to represent vectors, as in <q><m>\mathbf v</m></q>, or use arrows, as in <q><m>\oldvec v</m></q>.  As it is usually clear from context if a letter represents a vector, we do not decorate vectors in this way.</p>
    </remark>

    <note xml:id="vectors-diff-pts">
      <p>
      Another way to think about a vector is as a <em>difference</em> between two
    points, or the arrow from one point to another.  For instance, <m>{1\choose 2}</m> is the arrow from <m>(1,1)</m> to <m>(2,3)</m>.
    <latex-code>
          <![CDATA[
  \begin{tikzpicture}[whitebg nodes, thin border nodes]
  \draw[help lines] (0,0) grid (4,4);
  \draw[->] (0,0) -- (4,0);
  \draw[->] (0,0) -- (0,4);
  \point["${(1,1)}$" {left, text=seq-red}, seq-red] (x) at (1,1);
  \point["${(2,3)}$" {right, text=seq-green}, seq-green] (y) at (2,3);
  \draw[vector, seq-blue] (x)
    -- node[below right=1pt] {${1\choose 2}$} (y);
  \end{tikzpicture}
          ]]>
    </latex-code></p></note>

  </subsection>

  <subsection>
    <title>Vector Algebra and Geometry</title>

    <p>Here we learn how to add vectors together and how to multiply vectors by numbers, both algebraically and geometrically.</p>

    <definition hide-type="true">
      <title>Vector addition and scalar multiplication</title>
      <statement>
        <p><ul>
          <li>We can add two vectors together:
          <me>\vec{ a b c} + \vec{ x y z} = \vec{ a+x b+y c+z}.</me>
          <idx><h>Vector</h><h>addition</h></idx>
          </li>
          <li>We can multiply, or <term>scale</term>, a vector by a real number <m>c</m>:
          <idx><h>Vector</h><h>scalar multiplication</h></idx>
          <me>\def\r{\textcolor{red}}
\r c\vec{ x y z} = \vec{ \r c\cdot x \r c\cdot y \r c\cdot z}.</me>
          We call <m>c</m> a <term>scalar</term> to distinguish it from a vector.  If <m>v</m> is a vector and <m>c</m> is a scalar, then <m>cv</m> is called a <term>scalar multiple</term> of <m>v</m>.
          </li>
        </ul>
        Addition and scalar multiplication work in the same way for vectors of length <m>n</m>.</p>
      </statement>
    </definition>

    <example>
      <p><me>
        \vec{ 1 2 3} + \vec{ 4 5 6} = \vec{ 5 7 9} \sptxt{and}
        -2\vec{ 1 2 3} = \vec{-2 -4 -6}.
      </me></p>
    </example>

    <paragraphs>
      <title>The Parallelogram Law for Vector Addition</title>
      <p>
       Geometrically, the sum of two vectors <m>v,w</m> is obtained as follows: place the tail of <m>w</m> at the head of <m>v</m>. Then <m>v+w</m> is the vector whose tail is the tail of <m>v</m> and whose head is the head of <m>w</m>. Doing this both ways creates a parallelogram. For example,
       <me> \textcolor{seq-blue}{\vec{1 3}}
       + \textcolor{seq-green}{\vec{4 2}}
       = \vec{5 5}.
       </me>
       <idx><h>Vector</h><h>addition</h><h>parallelogram law</h></idx>
      </p>
      <p>Why? The width of <m>v+w</m> is the sum of the widths, and likewise with
       the heights.
       <latex-code>
            <![CDATA[
\begin{tikzpicture}
  \draw[grid lines] (0,0) grid (5,5);
  \fill[red!30, nearly transparent] (0,0) -- (1,3) -- (5,5) -- (4,2) -- cycle;
  \begin{scope}[thin border nodes]
  \draw[vector, seq-blue] (0,0) to["$v$"] (1,3);
  \draw[vector, seq-green] (1,3) to["$w$"] (5,5);
  \draw[vector, seq-green] (0,0) to["$w$"'] (4,2);
  \draw[vector, seq-blue] (4,2) to["$v$"'] (5,5);
  \draw[vector] (0,0) to["$v+w$" {sloped, pos=.6}] (5,5);
  \end{scope}
  \draw[|<->|] (0,5.5) -- (1,5.5);
  \draw[|<->|] (1,5.5) -- (5,5.5);
  \draw[|<->|] (0,-.5) -- (4,-.5);
  \draw[|<->|] (4,-.5) -- (5,-.5);
  \path (0,-.5) -- (5,-.5)
     node[pos=.5, font=\small, below=1mm] {$5 = 1 + 4 = 4+1$};
  \draw[|<->|] (5.5,0) -- (5.5,2);
  \draw[|<->|] (5.5,2) -- (5.5,5);
  \draw[|<->|] (-.5,0) -- (-.5,3);
  \draw[|<->|] (-.5,3) -- (-.5,5);
  \path (-.5, 0) -- (-.5, 5)
     node[pos=.5, font=\small, above, rotate=90] {$5 = 2 + 3 = 3 + 2$};
\end{tikzpicture}
            ]]>
      </latex-code></p>

      <example hide-type="true">
        <title>Interactive: The parallelogram law for vector addition</title>
        <figure>
          <caption>The parallelogram law for vector addition.  Click and drag the heads of <m></m> and <m>w</m>.</caption>
          <mathbox source="demos/vector-add.html" height="400px"/>
        </figure>
      </example>
    </paragraphs>

    <paragraphs>
      <title>Vector Subtraction</title>
       <idx><h>Vector</h><h>subtraction</h><h>picture of</h></idx>
      <p>Geometrically, the difference of two vectors <m>v,w</m> is obtained as follows: place the tail of <m>v</m> and <m>w</m> at the same point.  Then <m>v-w</m> is the vector from the head of <m>w</m> to the head of <m>v</m>. For example,
       <me> \textcolor{seq-blue}{\vec{1 4}}
       - \textcolor{seq-green}{\vec{4 2}}
       = \vec{-3 2}.
       </me>
      </p>
      <p>Why? If you add <m>v-w</m> to <m>w</m>, you get <m>v</m>.
      <latex-code>
            <![CDATA[
\begin{tikzpicture}
  \draw[grid lines] (0,0) grid (5,5);
  \begin{scope}[thin border nodes]
  \draw[vector, seq-blue] (0,0) to["$v$"] (1,4);
  \draw[vector, seq-green] (0,0) to["$w$"] (4,2);
  \draw[vector] (4,2) to["$v-w$"' {sloped, pos=.7}] (1,4);
  \end{scope}
\end{tikzpicture}
        ]]>
      </latex-code></p>

      <example hide-type="true">
        <title>Interactive: Vector subtraction</title>
        <figure>
          <caption>Vector subtraction.  Click and drag the heads of <m>v</m> and <m>w</m>.</caption>
          <mathbox source="demos/vector-sub.html" height="400px"/>
        </figure>
      </example>
    </paragraphs>

    <paragraphs>
      <title>Scalar Multiplication</title>

      <p>A scalar multiple of a vector <m>v</m> has the same (or opposite) direction, but a different length.  For instance, <m>2v</m> is the vector in the direction of <m>v</m> but twice as long, and <m>-\frac 12v</m> is the vector in the opposite direction of <m>v</m>, but half as long.  Note that the set of all scalar multiples of a (nonzero) vector <m>v</m> is a <em>line</em>.
      <idx><h>Vector</h><h>scalar multiplication</h><h>picture of</h></idx>
      <latex-code>
            <![CDATA[
  \begin{tikzpicture}
    \draw[grid lines] (-2,-2) grid (3,5);
    \node[anchor=south] at (.5,5) {Some multiples of $v$.};
    \draw[vector] (0,0) -- (1,2) node[below right, whitebg] {$v$};
    \draw[vector] (0,0) -- (2,4) node[below right, whitebg] {$2v$};
    \draw[vector] (0,0) -- (-.5,-1) node[left, whitebg] {$-\frac 12v$};
    \point["$0v$" below right] at (0,0);
    \begin{scope}[xshift=6cm]
    \node[anchor=south] at (.5,5) {All multiples of $v$.};
    \draw[help lines] (-2,-2) grid (3,5);
    \draw[thick, seq-blue, <->] (-1,-2) -- (2.5, 5);
    \point at (0,0);
    \end{scope}
  \end{tikzpicture}
        ]]>
      </latex-code></p>

      <example hide-type="true">
        <title>Interactive: Scalar multiplication</title>
        <figure>
          <caption>Scalar multiplication.  Drag the slider to change the scalar.</caption>
          <mathbox source="demos/vector-mul.html" height="400px"/>
        </figure>
      </example>

    </paragraphs>

  </subsection>

  <subsection>
    <title>Linear Combinations</title>

    <p>We can add and scale vectors in the same equation.</p>

    <definition>
      <idx><h>Linear combination</h><h>definition of</h></idx>
      <idx><h>Vector</h><h>linear combination of</h><see>Linear combination</see></idx>
      <statement>
        <p>Let <m>c_1,c_2,\ldots,c_k</m> be scalars, and let <m>v_1,v_2,\ldots,v_k</m> be vectors in <m>\R^n</m>.  The vector in <m>\R^n</m>
        <me>c_1v_1 + c_2v_2 + \cdots + c_kv_k
        </me>
        is called a <term>linear combination</term> of the vectors <m>v_1,v_2,\ldots,v_k</m>, with <term>weights</term> or <term>coefficients</term> <m>c_1,c_2,\ldots,c_k</m>.
        </p>
      </statement>
    </definition>

    <p>Geometrically, a linear combination is obtained by stretching / shrinking the vectors <m>v_1,v_2,\ldots,v_k</m> according to the coefficients, then adding them together using the parallelogram law.
    </p>

    <example xml:id="vectors-eg-lincombo-plane">
      <p>Let <m>v_1 = {1\choose 2}</m> and <m>v_2 = {1\choose 0}</m>.  Here are some linear combinations of <m>v_1</m> and <m>v_2</m>, drawn as points.
      <latex-code>
            <![CDATA[
  \begin{tikzpicture}[thin border nodes]
    \draw[help lines] (-3,-3) grid (4, 4);
    \draw[->] (-3,0) -- (4,0);
    \draw[->] (0,-3) -- (0,4);
    \draw[vector, seq1] (0,0) to["$v_1$"] (1,2);
    \draw[vector, seq2] (0,0) to["\strut$v_2$"'] (1,0);
    \point at (0,0);
    \point[seq3] at (2,2);
    \point[seq4] at (0,2);
    \point[seq5] at (2,4);
    \point[seq8] at (2,0);
    \point[seq7] at (-1,-2);

    \node[anchor=west, overlay] at (5, .75)
      {
        \begin{minipage}{0.6\linewidth}
           \begin{itemize}
           \color{seq3}
           \item $v_1+v_2$
           \color{seq4}
           \item $v_1-v_2$
           \color{seq5}
           \item $2v_1\color{gray}+0v_2$
           \color{seq8}
           \item $2v_2$
           \color{seq7}
           \item $-v_1$
           \end{itemize}
        \end{minipage}
      };
  \end{tikzpicture}
            ]]>
      </latex-code>
      <idx><h>Linear combination</h><h>two vectors, picture of</h></idx>
      The locations of these points are found using the parallelogram law for vector addition.  Any vector on the plane is a linear combination of <m>v_1</m> and <m>v_2</m>, with suitable coefficients.</p>
    </example>

    <figure>
      <caption>Linear combinations of two vectors in <m>\R^2</m>: move the sliders to change the coefficients of <m>v_1</m> and <m>v_2</m>.  Note that any vector on the plane can be obtained as a linear combination of <m>v_1,v_2</m> with suitable coefficients.</caption>
      <mathbox source="demos/spans.html?v1=1,2&amp;v2=1,0&amp;range=5&amp;captions=combo" height="400px"/>
    </figure>

    <example hide-type="true">
      <title>Interactive: Linear combinations of three vectors</title>
      <figure>
        <caption>Linear combinations of three vectors: move the sliders to change the coefficients of <m>v_1,v_2,v_3</m>.  Note how the parallelogram law for addition of three vectors is more of a <q>parallepiped law</q>.</caption>
        <mathbox source="demos/spans.html?v1=2,-1,1&amp;v2=1,1,-1&amp;v3=-1,1,4&amp;range=5&amp;captions=combo" height="500px"/>
      </figure>
    </example>

    <specialcase>
      <title>Linear Combinations of a Single Vector</title>
      <idx><h>Linear combination</h><h>single vector, picture of</h></idx>
      <p>A linear combination of a single vector <m>v = {1\choose 2}</m> is just a scalar multiple of <m>v</m>.  So some examples include
      <me>v=\vec{1 2},\quad \frac 32v=\vec{3/2 3},\quad -\frac12v = \vec{-1/2 -1},\quad\ldots
      </me>
      The set of all linear combinations is the <em>line through <m>v</m></em>.  (Unless <m>v=0</m>, in which case any scalar multiple of <m>v</m> is again <m>0</m>.)
      <latex-code>
            <![CDATA[
  \begin{tikzpicture}
    \draw[grid lines] (-3,-3) grid (4, 4);
    \clip (-3,-3) rectangle (4, 4);
    \draw[thin, seq4] ($-2*(2,1)$) -- ($2*(2,1)$);
    \draw[vector, seq1] (0,0) to["$v$" thin border] (2,1);
    \point at (0,0);
  \end{tikzpicture}
            ]]>
      </latex-code></p>
    </specialcase>

    <specialcase>
      <title>Linear Combinations of Collinear Vectors</title>
      <idx><h>Linear combination</h><h>collinear vectors, picture of</h></idx>
      <p>The set of all linear combinations of the vectors
      <me>v_1 = \vec{2 2} \sptxt{and} v_2 = \vec{-1 -1}</me>
      is the <em>line</em> containing both vectors.
      <latex-code>
            <![CDATA[
  \begin{tikzpicture}
    \draw[grid lines] (-3,-3) grid (4, 4);
    \path[clip] (-3,-3) rectangle (4, 4);
    \draw[thin, seq4] ($-2*(2,2)$) -- ($2*(2,2)$);
    \draw[vector, seq1] (0,0) to["$v_1$" thin border] (2,2);
    \draw[vector, seq2] (0,0) to["$v_2$"' thin border] (-1,-1);
    \point at (0,0);
  \end{tikzpicture}
            ]]>
      </latex-code>
      The difference between this and a previous <xref ref="vectors-eg-lincombo-plane">example</xref> is that both vectors lie on the same line.  Hence any scalar multiples of <m>v_1,v_2</m> lie on that line, as does their sum.
      </p>

    </specialcase>

    <example hide-type="true">
      <title>Interactive: Linear combinations of two collinear vectors</title>
      <figure>
        <caption>Linear combinations of two collinear vectors in <m>\R^2</m>.  Move the sliders to change the coefficients of <m>v_1,v_2</m>.  Note that there is no way to <q>escape</q> the line.</caption>
        <mathbox source="demos/spans.html?v1=2,2&amp;v2=-1,-1&amp;range=5&amp;captions=combo&amp;grid=disabled" height="400px"/>
      </figure>
    </example>

  </subsection>

</section>