1.1: Prelude to Analytic Geometry
- Page ID
- 3518
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In the \((x,y)\) coordinate system we normally write the \(x\)-axis horizontally, with positive numbers to the right of the origin, and the \(y\)-axis vertically, with positive numbers above the origin. That is, unless stated otherwise, we take "rightward'' to be the positive \(x\)-direction and "upward'' to be the positive \(y\)-direction. In a purely mathematical situation, we normally choose the same scale for the \(x\)- and \(y\)-axes. For example, the line joining the origin to the point \((a,a)\) makes an angle of 45\({}^\circ\) with the \(x\)-axis (and also with the \(y\)-axis).
In applications, often letters other than \(x\) and \(y\) are used, and often different scales are chosen in the horizontal and vertical directions. For example, suppose you drop something from a window, and you want to study how its height above the ground changes from second to second. It is natural to let the letter \(t\) denote the time (the number of seconds since the object was released) and to let the letter \(h\) denote the height. For each \(t\) (say, at one-second intervals) you have a corresponding height \(h\). This information can be tabulated, and then plotted on the \((t,h)\) coordinate plane, as shown in figure 1.0.1.
| seconds | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| meters | 80 | 75.1 | 60.4 | 35.9 | 1.6 |
We use the word "quadrant'' for each of the four regions into which the plane is divided by the axes: the first quadrant is where points have both coordinates positive, or the "northeast'' portion of the plot, and the second, third, and fourth quadrants are counted off counterclockwise, so the second quadrant is the northwest, the third is the southwest, and the fourth is the southeast.
Suppose we have two points \(A\) and \(B\) in the \((x,y)\)-plane. We often want to know the change in \(x\)-coordinate (also called the "horizontal distance'') in going from \(A\) to \(B\). This is often written \(\Delta x\), where the meaning of \(\Delta\) (a capital delta in the Greek alphabet) is "change in''. (Thus, \(\Delta x\) can be read as "change in \(x\)'' although it usually is read as "delta \(x\)''. The point is that \(\Delta x\) denotes a single number, and should not be interpreted as "delta times \(x\)''.) For example, if \(A=(2,1)\) and \(B=(3,3)\), \(\Delta x=3-2=1\). Similarly, the "change in \(y\)'' is written \(\Delta y\). In our example, \(\Delta y= 3-1=2\), the difference between the \(y\)-coordinates of the two points. It is the vertical distance you have to move in going from \(A\) to \(B\). The general formulas for the change in \(x\) and the change in \(y\) between a point \((x_1,y_1)\) and a point \((x_2,y_2)\) are: $$ \Delta x=x_2-x_1,\qquad\qquad\Delta y=y_2-y_1. $$ Note that either or both of these might be negative.
Contributors
Integrated by Justin Marshall.