A city planner in Chicago wants a new street to meet Lake Shore Drive at a perfect right angle. A drone team needs to know exactly how two flight paths cross. On a coordinate map you don't need a protractor: the equations of the lines already tell you the angle. In this section you will learn how to read that angle from slopes or direction vectors, and how to build lines that are parallel or perpendicular to a given one.
Slope and angle of inclination
Every non-vertical line leans at some angle. The angle of inclination $\theta$ is the angle measured counterclockwise from the positive x-axis up to the line, with $0^\circ \le \theta < 180^\circ$.
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Determine the angle that the line $\sqrt{3}x - y + 2 = 0$ forms with the positive direction of the x-axis.
Determine the angle that the line $\sqrt{3}x - y + 2 = 0$ forms with the positive direction of the x-axis.
Two lines that cross make four angles: two equal acute ones and two equal obtuse ones. By convention, "the angle between two lines" means the acute one (or $90^\circ$ if they are perpendicular).
Why does a formula exist? If $r$ and $s$ have inclinations $\theta_1$ and $\theta_2$, the angle between them is the difference $\theta_2 - \theta_1$. The tangent difference identity gives
Two lines that lean the same way never meet. The slope measures the lean, so equal slopes means parallel.
Many problems call $Ax + By + C = 0$ the general equation. In US textbooks it is standard form with every term moved to one side, and we will call it general form here. A handy shortcut: a line parallel to $Ax + By + C = 0$ has the same $A$ and $B$ and only a different constant: $Ax + By + C' = 0$. Then you plug in the given point to find $C'$.
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Determine the general equation of the line parallel to $y = 3x - 5$ that passes through the point $P(2, -1)$.
Determine the general equation of the line parallel to $y = 3x - 5$ that passes through the point $P(2, -1)$.
Rotating a direction $\langle a, b\rangle$ by $90^\circ$ gives $\langle -b, a\rangle$: run and rise swap places and one changes sign. That is why the slope of a perpendicular line is the negative reciprocal.
Drag the slider: whatever the slope $m$, the second line stays at a right angle to the first.
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Find the equation of the line perpendicular to $x + 4y - 7 = 0$ that passes through the origin. Express the result in slope-intercept form.
Find the equation of the line perpendicular to $x + 4y - 7 = 0$ that passes through the origin. Express the result in slope-intercept form.
Determine which of the following lines are parallel and which are perpendicular to each other: $r_1: 2x - 3y + 1 = 0$ $r_2: 6x + 4y - 5 = 0$ $r_3: -4x + 6y + 7 = 0$
Determine which of the following lines are parallel and which are perpendicular to each other:
Exact angle $\alpha$: set $\left|\dfrac{ab' - a'b}{aa' + bb'}\right| = \tan\alpha$ and solve.
Prefer the general-form version for the angle condition: a parameter can make a line vertical (for example $k = 0$ in $2x + ky - 5 = 0$), and the slope formula would miss that case.
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Calculate the values of the parameter $k$ so that the line $kx + y - 2 = 0$ forms an angle of $45^\circ$ with the line of equation $x - 2y + 1 = 0$.
Calculate the values of the parameter $k$ so that the line $kx + y - 2 = 0$ forms an angle of $45^\circ$ with the line of equation $x - 2y + 1 = 0$.
Two airplanes follow rectilinear trajectories on a radar screen given by the equations $r: x - 3y + 2 = 0$ and $s: 2x + ky - 5 = 0$. Determine the value (or values) of…
Two airplanes follow rectilinear trajectories on a radar screen given by the equations $r: x - 3y + 2 = 0$ and $s: 2x + ky - 5 = 0$. Determine the value (or values) of $k$ so that their routes cross forming an exact angle of $60^\circ$.
Two robotic arms in a factory move on rectilinear rails. The first follows the line $r: 3x + 4y - 12 = 0$ and the second the line $s: ax + 8y - 5 = 0$. 1. Determine $a$…
Two robotic arms in a factory move on rectilinear rails. The first follows the line $r: 3x + 4y - 12 = 0$ and the second the line $s: ax + 8y - 5 = 0$.
Determine $a$ so that the rails are parallel.
Find the equation of a third transverse rail that is perpendicular to both and crosses rail $r$ exactly at its point of intersection with the y-axis.
An equation like $(2 + k)x + (2k - 1)y - 5k = 0$ describes a different line for each value of $k$. Group the terms with $k$:
$$(2x - y) + k(x + 2y - 5) = 0.$$
If a point satisfies both $2x - y = 0$ and $x + 2y - 5 = 0$, then the equation holds for every $k$. So every line of the family passes through the intersection of those two lines.
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Given the family of lines depending on the parameter $k \in \mathbb{R}$: $(2 + k)x + (1 - 2k)y + 3k - 4 = 0$. Prove that all the lines in this family pass through the…
Given the family of lines depending on the parameter $k \in \mathbb{R}$: $(2 + k)x + (1 - 2k)y + 3k - 4 = 0$. Prove that all the lines in this family pass through the same fixed point and calculate its coordinates. Then, find the value of $k$ to obtain the line in the family that is perpendicular to $3x - y + 2 = 0$.
by splitting it into $m - m_0 = \pm\tan\alpha\,(1 + m\,m_0)$. You normally get two slopes, one on each side of the given line. Then use point-slope form with $P$.
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A vertex of an equilateral triangle is the origin $(0,0)$ and the opposite side is contained in the line $x + y - 4 = 0$. Determine the general equations of the lines…
A vertex of an equilateral triangle is the origin $(0,0)$ and the opposite side is contained in the line $x + y - 4 = 0$. Determine the general equations of the lines containing the other two sides of the triangle knowing that they must form an angle of $60^\circ$ with the given side.
A solar panel is aligned with the line $r: \sqrt{3}x - y + 10 = 0$. The sun's rays fall parallel to the line $s: 2x + 2y - 5 = 0$. Calculate the angle of incidence of…
A solar panel is aligned with the line $r: \sqrt{3}x - y + 10 = 0$. The sun's rays fall parallel to the line $s: 2x + 2y - 5 = 0$. Calculate the angle of incidence of the rays on the panel. If the panel must be reoriented by pivoting on its point of intersection with the $OY$-axis so that the rays fall completely perpendicular to its surface, what will be the new general equation of the line that defines the panel?
Prove that if two lines $r: ax + by + c = 0$ and $s: a'x + b'y + c' = 0$ intersect forming an angle $\alpha$, then…
Prove that if two lines $r: ax + by + c = 0$ and $s: a'x + b'y + c' = 0$ intersect forming an angle $\alpha$, then $\tan \alpha = \left| \frac{ab' - a'b}{aa' + bb'} \right|$ is satisfied. Use this result to find the general equations of the lines that pass through $P(1,1)$ and form an angle $\alpha$ such that $\tan \alpha = \frac{1}{2}$ with the line $2x - y + 3 = 0$.
Perpendicular lines are the key to two classic constructions.
Recipe: (1) find the slope of the side, (2) take the negative reciprocal, (3) use point-slope form with the vertex (altitude) or the midpoint (perpendicular bisector). If the problem gives a point as the intersection of two lines, solve that system first to get the coordinates.
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In the design of a park, a new avenue must be perpendicular to the main street defined by the equation $3x - 4y + 10 = 0$. Furthermore, this avenue must pass through the…
In the design of a park, a new avenue must be perpendicular to the main street defined by the equation $3x - 4y + 10 = 0$. Furthermore, this avenue must pass through the point of intersection of the secondary streets $x + y - 2 = 0$ and $2x - y - 4 = 0$. Find the general equation of the new avenue.
Given a triangle with vertices $A(1, 2)$, $B(5, 3)$ and $C(4, 6)$, find the general equation of the altitude corresponding to the vertex $A$ and calculate the angle this…
Given a triangle with vertices $A(1, 2)$, $B(5, 3)$ and $C(4, 6)$, find the general equation of the altitude corresponding to the vertex $A$ and calculate the angle this altitude forms with the side $AB$.
On the plan of a metal structure, a main support beam rests on the line passing through the nodes $P(-2, 4)$ and $Q(4, 1)$. A second beam needs to be welded…
On the plan of a metal structure, a main support beam rests on the line passing through the nodes $P(-2, 4)$ and $Q(4, 1)$. A second beam needs to be welded perpendicular to the first exactly at its midpoint. Find the slope-intercept equation of the line that defines this second beam.
The orthogonal projection of a point $P$ onto a line $L$ is the foot $F$ of the perpendicular from $P$ to $L$. The reflection $P'$ of $P$ in $L$ is the point on the other side with $F$ as the midpoint of $PP'$, so $P' = 2F - P$ (coordinate by coordinate).
To reflect a line $r$ in a line $s$: (1) find the intersection $Q$ of $r$ and $s$ (it does not move); (2) pick another point of $r$ and reflect it as above; (3) draw the line through $Q$ and that image.
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A laser beam travels along the line $y = -2x + 5$ and reflects off a plane mirror located on the x-axis. Calculate the angle of incidence (angle between the beam and the…
A laser beam travels along the line $y = -2x + 5$ and reflects off a plane mirror located on the x-axis. Calculate the angle of incidence (angle between the beam and the normal to the mirror) and determine the equation of the line containing the reflected beam.
Two crossing lines form two pairs of vertical angles, so there are two angle bisectors. Every point on a bisector is equally far from both lines.
If the two normal lengths differ, as in $3x - 4y + 5 = 0$ (length $5$) and $12x + 5y - 1 = 0$ (length $13$), cross-multiply: $13(3x - 4y + 5) = \pm 5(12x + 5y - 1)$.
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Find the equations of the angle bisectors of the angles formed by the lines $r: 3x - 4y + 5 = 0$ and $s: 12x + 5y - 1 = 0$. Prove analytically, using their direction…
Find the equations of the angle bisectors of the angles formed by the lines $r: 3x - 4y + 5 = 0$ and $s: 12x + 5y - 1 = 0$. Prove analytically, using their direction vectors or their slopes, that both angle bisectors are perpendicular to each other.
Coordinates plus perpendicular lines let you rebuild a polygon from partial data. Two facts do most of the work.
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The diagonals of a rhombus are contained in the lines $r: x + 2y - 4 = 0$ and $s: 2x - y - 3 = 0$. Knowing that one of its vertices is $A(0, 2)$ and that the length of…
The diagonals of a rhombus are contained in the lines $r: x + 2y - 4 = 0$ and $s: 2x - y - 3 = 0$. Knowing that one of its vertices is $A(0, 2)$ and that the length of its side is $\sqrt{20}$, find the coordinates of the other three vertices.
Two opposite vertices of a square are $A(-1, 3)$ and $C(5, 1)$. Find the equations of the lines containing the four sides of the square, first calculating its center and…
Two opposite vertices of a square are $A(-1, 3)$ and $C(5, 1)$. Find the equations of the lines containing the four sides of the square, first calculating its center and the directions of its diagonals.
Back to the start: to check that a new street is perpendicular to a road with slope $m$, test whether the product of the slopes is $-1$, or build the street with slope $-\dfrac1m$ through the point where it must start. No protractor needed.
Let's practice! 8 medium exercises in this section