| Version 7 |
Version 6 |
| \subsubsection*{Equation of a line} |
\subsubsection*{Equation of a line} |
| Suppose $a,b,c\in \R$. Then the set of points $(x,y)$ in the |
Suppose $a,b,c\in \R$. Then the set of points $(x,y)$ in the |
| plane that satisf{y} |
plane that satisf{y} |
| $$ |
$$ |
| ax + by = c, |
ax + by = c, |
| $$ |
$$ |
| where $a$ and $b$ can not be both 0, is an (infinite) \emph{line}. |
where $a$ and $b$ can not be both 0, is an (infinite) \emph{line}. |
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| The value of $y$ when $x=0$, if it exists, is called the \emph{$y$-intercept}. Geometrically, if $d$ is the $y$-intercept, then $(0,d)$ is the point of intersection of the line and the $y$-axis. The $y$-intercept exists iff the line is not parallel to the $y$-axis. The \emph{$x$-intercept} is defined similarly. |
The value of $y$ when $x=0$, if it exists, is called the \emph{$y$-intercept}. Geometrically, if $d$ is the $y$-intercept, then $(0,d)$ is the point of intersection of the line and the $y$-axis. The $y$-intercept exists iff the line is not parallel to the $y$-axis. The \emph{$x$-intercept} is defined similarly. |
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| If $b\neq0$, then the above equation of the line can be rewritten as |
If $b\neq0$, then the above equation of the line can be rewritten as |
| $$ |
$$ |
| y = mx + d. |
y = mx + d. |
| $$ |
$$ |
| This is called the \emph{slope-intercept form} of a line, because both the slope and the $y$-intercept are easily identifiable in the equation. The slope is $m$ and the $y$-intercept is $d$. |
This is called the \emph{slope-intercept form} of a line, because both the slope and the $y$-intercept are easily identifiable in the equation. The slope is $m$ and the $y$-intercept is $d$. |
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| Three finite points $(x_i, y_i),\; i=1,2,3$ in $\R^2$ are colinear if and only if the following determinant vanishes: |
Three finite points $(x_i, y_i),\; i=1,2,3$ in $\R^2$ are colinear if and only if the following determinant vanishes: |
| $$\left| \begin{array}{ccc} x_1 & x_2 &x_3 \\ y_1 & y_2 & y_3 \\ 1 & 1& 1\end{array} \right|=0.$$ |
$$\left| \begin{array}{ccc} x_1 & x_2 &x_3 \\ y_1 & y_2 & y_3 \\ 1 & 1& 1\end{array} \right|=0.$$ |
| Therefore, the line between distinct points $(x_1,y_1), (x_2,y_2)$ has equation |
Therefore, the line between distinct points $(x_1,y_1), (x_2,y_2)$ has equation |
| $$\left| \begin{array}{ccc} x_1 & x_2 &x \\ y_1 & y_2 & y \\ 1 & 1& 1\end{array} \right|=0,$$ |
$$\left| \begin{array}{ccc} x_1 & x_2 &x \\ y_1 & y_2 & y \\ 1 & 1& 1\end{array} \right|=0,$$ |
| or more simply |
or more simply |
| $$ |
$$ |
| (y_1-y_2)x+(x_2 - x_1)y + y_2 x_1-x_2 y_1=0. |
(y_1-y_2)x+(x_2 - x_1)y + y_2 x_1-x_2 y_1=0. |
| $$ |
$$ |
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| \subsubsection*{Line segment} |
\subsubsection*{Line segment} |
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| Let $p_1=(x_1,y_1), p_2=(x_2,y_2)$ be distinct points in $\R^2$. The closed line segement generated by these points is the set |
Let $p_1=(x_1,y_1), p_2=(x_2,y_2)$ be distinct points in $\R^2$. The closed line segement generated by these points is the set |
| $$\{ p\in \R^2 \mid p=t p_1+(1-t) p_2,\; 0\leq t\leq 1\}.$$ |
$$\{ p\in \R^2 \mid p=t p_1+(1-t) p_2,\; 0\leq t\leq 1\}.$$ |
