Preview Activity 2.1.1 Scalar Multiplication and Vector Addition

Suppose that

\begin{equation*} \vvec = \left[\begin{array}{r} 3 \\ 1 \end{array} \right], \wvec = \left[\begin{array}{r} -1 \\ 2 \end{array} \right]. \end{equation*}
  1. Find expressions for the vectors

    \begin{equation*} \begin{array}{cccc} \vvec, \amp 2\vvec, \amp -\vvec, \amp -2\vvec, \\ \wvec, \amp 2\wvec, \amp -\wvec, \amp -2\wvec\text{.} \\ \end{array} \end{equation*}

    and sketch them below.

  2. What geometric effect does scalar multiplication have on a vector? Also, describe the effect multiplying by a negative scalar has.

  3. Sketch the vectors \(\vvec, \wvec, \vvec + \wvec\) below.

  4. Consider vectors that have the form \(\vvec + a\wvec\) where \(a\) is any scalar. Sketch a few of these vectors when, say, \(a = -2, -1, 0, 1, \) and \(2\text{.}\) Give a geometric description of this set of vectors.

  5. If \(a\) and \(b\) are two scalars, then the vector

    \begin{equation*} a \vvec + b \wvec \end{equation*}

    is called a linear combination of the vectors \(\vvec\) and \(\wvec\text{.}\) Find the vector that is the linear combination when \(a = -2\) and \(b = 1\text{.}\)

  6. Can the vector \(\left[\begin{array}{r} -31 \\ 37 \end{array}\right]\) be represented as a linear combination of \(\vvec\) and \(\wvec\text{?}\)