Preview Activity2.1.1Scalar 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*}
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.
What geometric effect does scalar multiplication have on a vector? Also, describe the effect multiplying by a negative scalar has.
Sketch the vectors \(\vvec, \wvec, \vvec + \wvec\) below.
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.

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 \(b = 1\text{.}\)
Can the vector \(\left[\begin{array}{r} 31 \\ 37 \end{array}\right]\) be represented as a linear combination of \(\vvec\) and \(\wvec\text{?}\)