Preview Activity 9.5.1.
We will start our work on lines by considering some familiar ideas in \(\R^2\) but from a new perspective. You are probably familiar with equations of lines in the \(xy\)-plane in the form \(y = mx+b\text{,}\) where \(m\) is the slope of the line and \((0,b)\) is the \(y\)-intercept. In this activity, we explore a more flexible way of representing lines that is useful in the \(xy\)-plane and higher dimensions. To begin, consider the line through the point \((2,-1)\) with slope \(\frac{2}{3}\) as shown in Figure 9.5.2.
A graph with axes showing the point \((2,-1)\) and the line with slope \(2/3\) passing through that point.
(a)
Suppose we increase \(x\) by 1 from the point \((2,-1)\text{.}\) How does the \(y\)-value change? What is the point on the line with \(x\)-coordinate \(3\text{?}\)
(b)
Suppose we decrease \(x\) by 3.25 from the point \((2,-1)\text{.}\) How does the \(y\)-value change? What is the point on the line with \(x\)-coordinate \(-1.25\text{?}\)
(c)
Now, suppose we increase \(x\) by some arbitrary value \(3t\) from the point \((2,-1)\text{.}\) How does the \(y\)-value change? What is the point on the line with \(x\)-coordinate \(2+3t\text{?}\)
(d)
Remember that the horizontal component of a vector describes the “run” and the vertical component measures the “rise” of the vector. So the slope of the line is related to any vector whose \(y\)-component divided by the \(x\)-component is the slope of the line. For the line in this activity, we might use the vector \(\langle 3,2 \rangle\) to describe the direction of the line. We will look at the following vector-valued function of one variable:
\begin{equation*}
\vr(t) = \langle 2,-1 \rangle + \langle 3,2 \rangle t,
\end{equation*}
For each of the values of \(t\) below, find \(\vr(t)\) and write your result as a single vector (of the form \(\langle a,b\rangle\))
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\(\displaystyle t=0\)
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\(\displaystyle t=1\)
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\(\displaystyle t=2\)
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\(\displaystyle t=-2\)
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\(\displaystyle t=-\frac{1}{2}\)
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\(\displaystyle t=\frac{7}{3}\)
(e)
Draw the six vectors from the previous step in standard position (with initial point at the origin) on the plot below.
A blank 2D plot where each coordinate varies from negative 10 to 10
(f)
Write a few sentences that compare the endpoints of your vectors from the previous task to the plot of the line through the point \((2, -1)\) with a slope of \(\frac{2}{3}\) (Figure 9.5.2). In particular you should address what aspects of the line are related to \(\langle 2, -1 \rangle\) or \(\langle 3,2 \rangle\text{.}\)
