Horizontal and Vertical Lines
Picture a city map laid out as a grid, like downtown Chicago. Walk east along one street and your north-south position never changes. Walk north along an avenue and your east-west position never changes. Those "never changes" paths are the simplest lines in the coordinate plane, and you can write each one with a single short equation. How? Let's find out.
Horizontal lines
Why does this work? Moving along a horizontal line, you never go up or down, so $y$ stays the same while $x$ takes any value.
Use the slider to move a horizontal line up and down. Only $y$ changes; the line stays flat.
Write down the equation of the line parallel to the x-axis ($X$-axis) that passes through the point $A(3, -5)$.
Write down the equation of the line parallel to the x-axis ($X$-axis) that passes through the point $A(3, -5)$.
Vertical lines and the axes
Moving along a vertical line you go straight up or down, so $x$ never changes while $y$ takes any value.
The two coordinate axes are special cases of these lines:
| Line | Equation |
|---|---|
| x-axis | $y = 0$ |
| y-axis | $x = 0$ |
Write down the equation of the line parallel to the y-axis ($Y$-axis) that passes through the point $B(-2, 7)$.
Write down the equation of the line parallel to the y-axis ($Y$-axis) that passes through the point $B(-2, 7)$.
Write down the general equations of the coordinate axes $X$ and $Y$.
Write down the general equations of the coordinate axes $X$ and $Y$.
A delivery drone moves through a city with grid-like streets. Its path follows a straight line parallel to the $Y$-axis passing through the coordinate $(6, 2)$. Write…
A delivery drone moves through a city with grid-like streets. Its path follows a straight line parallel to the $Y$-axis passing through the coordinate $(6, 2)$. Write down the equation of its path and the coordinates of two points it will pass through.
Intersections and perpendicular lines
A vertical line $x = a$ and a horizontal line $y = b$ always cross at exactly one point: $(a, b)$. They also meet at a right angle, so a vertical line is perpendicular to every horizontal line.
To find where a horizontal or vertical line meets another line, substitute the known value into the other equation.
To find the line perpendicular to a given horizontal or vertical line, just flip the type. The perpendicular to $y = 5$ through $(2, 9)$ is vertical, so it is $x = 2$.
Determine the coordinates of the point of intersection of the lines $r: x = 5$ and $s: y = -2$.
Determine the coordinates of the point of intersection of the lines $r: x = 5$ and $s: y = -2$.
Find the equation of the line perpendicular to $x = 7$ that passes through the point $M(-4, 9)$.
Find the equation of the line perpendicular to $x = 7$ that passes through the point $M(-4, 9)$.
An electronic components factory has fixed monthly costs of $12000\text{ \$}$, regardless of the quantity $x$ of parts manufactured. The revenue is given by the line $y…
An electronic components factory has fixed monthly costs of $12000\text{ \$}$, regardless of the quantity $x$ of parts manufactured. The revenue is given by the line $y = 30x$. Write the equation of the fixed cost line on the $(x, y)$ plane and find the exact point where the revenue equals the fixed costs.
Slope and direction vector
A direction vector $\langle a, b\rangle$ of a line points along the line: moving $a$ units right and $b$ units up keeps you on it. The slope is then $m = \dfrac{b}{a}$, which is "rise over run".
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A horizontal line has direction vector $\langle 1, 0\rangle$: run $1$, rise $0$. So $m = \dfrac{0}{1} = 0$.
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A vertical line has direction vector $\langle 0, 1\rangle$: run $0$, rise $1$. The slope $\dfrac{1}{0}$ is undefined, because you can't divide by $0$.
Any nonzero multiple works too. For example, $\langle 5, 0\rangle$ is also a direction vector of a horizontal line. Some problems call the slope the gradient; it is the same number.
State the direction vector and the gradient of the line with equation $y = 4$.
State the direction vector and the gradient of the line with equation $y = 4$.
State the direction vector and the gradient of the line with equation $x = -1$.
State the direction vector and the gradient of the line with equation $x = -1$.
In a chemistry experiment, the temperature $T$ (in degrees Fahrenheit) is kept constant at $35^\circ\text{C}$ over time $t$ (in minutes). Representing time on the…
In a chemistry experiment, the temperature $T$ (in degrees Fahrenheit) is kept constant at $35^\circ\text{C}$ over time $t$ (in minutes). Representing time on the $X$-axis and temperature on the $Y$-axis, write down the equation of the line and its direction vector.
Parametric and general equations
A line can also be described by a starting point $(x_0, y_0)$ and a direction vector $\langle a, b\rangle$:
$$x = x_0 + at, \qquad y = y_0 + bt$$As the parameter $t$ changes, the point $(x, y)$ slides along the line.
For lines parallel to the axes this gets very simple:
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Horizontal line $y = k$: $x = t,\; y = k$. (Direction vector $\langle 1, 0\rangle$.)
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Vertical line $x = k$: $x = k,\; y = t$. (Direction vector $\langle 0, 1\rangle$.)
The general equation puts everything on one side: $Ax + By + C = 0$. It is the same idea as standard form $Ax + By = C$. For example, $y = 2$ becomes $y - 2 = 0$, and $x = -3$ becomes $x + 3 = 0$.
Given the line in parametric form $x = -3$, $y = t$, state which axis it is parallel to and write down its general equation.
Given the line in parametric form $x = -3$, $y = t$, state which axis it is parallel to and write down its general equation.
The printhead of a 3D printer must draw a constant horizontal support line at $15\text{ mm}$ from the origin on the $Y$-axis. Write down the parametric equations and the…
The printhead of a 3D printer must draw a constant horizontal support line at $15\text{ mm}$ from the origin on the $Y$-axis. Write down the parametric equations and the general equation of this line.
Recognizing them from two points
Drag the point below to $(-3, 2)$ and then to $(4, 2)$. What stays the same?
A stepped path (two segments, one parallel to each axis) is just one horizontal line and one vertical line.
Altitudes. An altitude of a triangle is perpendicular to a side. If that side is horizontal, the altitude is vertical, and the other way around.
A line passes through the points $P(4, 2)$ and $Q(-3, 2)$. Is it parallel to either of the axes? Write down its equation.
A line passes through the points $P(4, 2)$ and $Q(-3, 2)$. Is it parallel to either of the axes? Write down its equation.
A robotic arm must move from the point $P(-2, -5)$ to the point $Q(4, 7)$ following a stepped path composed of two segments: first a movement parallel to the $X$-axis…
A robotic arm must move from the point $P(-2, -5)$ to the point $Q(4, 7)$ following a stepped path composed of two segments: first a movement parallel to the $X$-axis and then a movement parallel to the $Y$-axis. Write the equations of the lines that contain these two segments.
Given the triangle with vertices $A(-3, 2)$, $B(5, 2)$ and $C(1, 8)$, find the equation of the altitude from vertex $C$. Which axis is this altitude parallel to?
Given the triangle with vertices $A(-3, 2)$, $B(5, 2)$ and $C(1, 8)$, find the equation of the altitude from vertex $C$. Which axis is this altitude parallel to?
Spotting them in general form
In $Ax + By + C = 0$, a line parallel to an axis is missing one variable:
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If $B = 0$ (no $y$ term), then $Ax + C = 0$, so $x = -\dfrac{C}{A}$. The line is vertical.
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If $A = 0$ (no $x$ term), then $By + C = 0$, so $y = -\dfrac{C}{B}$. The line is horizontal.
In both cases, the coefficients $A$ and $B$ can't both be $0$, or there is no line at all.
When both coefficients depend on a parameter, check that they don't vanish together.
Determine the value of $k$ so that the line with general equation $kx + 3y - 6 = 0$ is parallel to the $X$-axis. Write down the resulting equation.
Determine the value of $k$ so that the line with general equation $kx + 3y - 6 = 0$ is parallel to the $X$-axis. Write down the resulting equation.
Determine the value of $a$ so that the line with equation $4x + (a+1)y - 8 = 0$ is parallel to the $Y$-axis. Write down the resulting equation.
Determine the value of $a$ so that the line with equation $4x + (a+1)y - 8 = 0$ is parallel to the $Y$-axis. Write down the resulting equation.
Find the value of $m$ so that the line $(m^2-9)x + (m+3)y + 5 = 0$ is parallel to the $X$-axis. Is there any value of $m$ for which this same line is parallel to the…
Find the value of $m$ so that the line $(m^2-9)x + (m+3)y + 5 = 0$ is parallel to the $X$-axis. Is there any value of $m$ for which this same line is parallel to the $Y$-axis? Justify your answer.
Distances and relative position
Relative position of two lines. When one of the lines is $y = k$ or $x = h$, substitute it into the other equation. The result tells you whether the lines are parallel, coincident (the same line), or intersecting.
Calculate the minimum distance separating the parallel lines $r: y = 8$ and $s: y = -3$.
Calculate the minimum distance separating the parallel lines $r: y = 8$ and $s: y = -3$.
A drip irrigation system is installed along a pipe parallel to the $Y$-axis. We know that the pipe must water a tree located at the point $A(5, 12)$ and that the exact…
A drip irrigation system is installed along a pipe parallel to the $Y$-axis. We know that the pipe must water a tree located at the point $A(5, 12)$ and that the exact distance from the pipe to the $Y$-axis is $5\text{ m}$. Write down the equation of the line representing the pipe.
Determine the relative position of the line $r: x = 4$ and the line $s: 2x + ky - 8 = 0$ depending on the parameter $k$.
Determine the relative position of the line $r: x = 4$ and the line $s: 2x + ky - 8 = 0$ depending on the parameter $k$.
Rectangles aligned with the axes
The sides of any rectangle whose sides are parallel to the axes lie on two horizontal and two vertical lines. Basketball courts, phone screens, and floor plans are full of them.
Back to the city grid: walking along a street keeps your north-south position fixed, which is a horizontal line $y = k$. Walking along an avenue keeps your east-west position fixed, which is a vertical line $x = h$. Where a street and an avenue cross, you are at the point $(h, k)$.
On a soccer field, we represent the center of the field at the origin $(0,0)$. If the field is $100\text{ m}$ long (located on the $X$-axis) and $60\text{ m}$ wide…
On a soccer field, we represent the center of the field at the origin $(0,0)$. If the field is $100\text{ m}$ long (located on the $X$-axis) and $60\text{ m}$ wide (located on the $Y$-axis), write down the equations of the four straight lines that bound the playing field.
In the development of a video game, the screen has a resolution of $1920 \times 1080$ pixels. If we place the origin $(0,0)$ at the bottom-left corner, write the…
In the development of a video game, the screen has a resolution of $1920 \times 1080$ pixels. If we place the origin $(0,0)$ at the bottom-left corner, write the equations of the two lines (one horizontal and one vertical) that divide the screen exactly into four equal quadrants.
A square has its center at the point $C(3, 3)$ and its sides are parallel to the coordinate axes. If the side of the square measures $4\text{ u}$, write the equations of…
A square has its center at the point $C(3, 3)$ and its sides are parallel to the coordinate axes. If the side of the square measures $4\text{ u}$, write the equations of the four lines that contain its sides.