Skip to main content Contents Prev Up Next \(\newcommand{\avec}{{\mathbf a}}
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Section 1.1 Scalar Multiplication
Activity 1.1.1 . 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*}
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 using
Figure 1.1.1 .
Figure 1.1.1. Sketch the vectors on this grid.
What geometric effect does scalar multiplication have on a vector? Also, describe the effect that multiplying by a negative scalar has.
Sketch the vectors
\(\vvec, \wvec, \vvec + \wvec\) using
Figure 1.1.2 .
Figure 1.1.2. Sketch the vectors on this grid.
Consider vectors that have the form
\(\vvec +
c\wvec\) where
\(c\) is any scalar. Sketch a few of these vectors when, say,
\(c = -2, -1, 0, 1, \) and
\(2\text{.}\) Give a geometric description of this set of vectors.
Figure 1.1.3. Sketch the vectors on this grid.
If \(c\) and \(d\) are two scalars, then the vector
\begin{equation*}
c \vvec + d \wvec
\end{equation*}
is called a linear combination of the vectors \(\vvec\) and \(\wvec\text{.}\) Find the vector that is the linear combination when \(c = -2\) and \(d = 1\text{.}\)
Solution . Solutions to this preview activity are given in the text below.
