##### 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*}

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 $$\wvec\text{?}$$

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