Angle Between Two Lines. Parallelism and Perpendicularity
At a factory in Monterrey, two robotic arms move along straight rails. To keep them from colliding, the engineer needs to know whether the rails are parallel, whether they cross at a right angle, or what angle they form. No protractor is needed: the equations of the lines are enough. On this page you will learn how to figure it out.
The Angle Between Two Lines
Two lines that intersect form four angles: two equal acute angles and two equal obtuse angles. By convention, the angle between two lines is the acute one (from $0^\circ$ to $90^\circ$).
Where does it come from? Each slope is the tangent of the line's angle of inclination ($m=\tan\theta$). The angle between the lines is the difference of the inclinations, $\alpha=\theta_1-\theta_2$, and the formula is the one for the tangent of a difference. The absolute value bars guarantee that we get the acute angle.
Calculate the angle formed by the lines given by their explicit equations $y = 2x + 1$ and $y = -3x + 4$.
Calculate the angle formed by the lines given by their explicit equations $y = 2x + 1$ and $y = -3x + 4$.
Calculate the acute angle between the lines $r: 2x - y + 4 = 0$ and $s: 3x + y - 1 = 0$.
Calculate the acute angle between the lines $r: 2x - y + 4 = 0$ and $s: 3x + y - 1 = 0$.
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.
Parallel Lines
Determine the general equation of the line parallel to $y = 3x - 5$ passing through the point $P(2, -1)$.
Determine the general equation of the line parallel to $y = 3x - 5$ passing through the point $P(2, -1)$.
Given the lines $r: kx - 2y + 5 = 0$ and $s: 3x + 4y - 1 = 0$, find the value of the parameter $k$ so that both lines are parallel.
Given the lines $r: kx - 2y + 5 = 0$ and $s: 3x + 4y - 1 = 0$, find the value of the parameter $k$ so that both lines are parallel.
Perpendicular Lines
Find the equation of the line perpendicular to $x + 4y - 7 = 0$ passing through the origin of coordinates. Express the result in explicit form.
Find the equation of the line perpendicular to $x + 4y - 7 = 0$ passing through the origin of coordinates. Express the result in explicit form.
Find the value of $m$ so that the lines $r: 5x + 2y = 0$ and $s: mx - 5y + 3 = 0$ are perpendicular.
Find the value of $m$ so that the lines $r: 5x + 2y = 0$ and $s: mx - 5y + 3 = 0$ are perpendicular.
State which of the following lines are parallel and which are perpendicular to each other: $r_1: 2x - 3y + 1 = 0$ $r_2: 6x + 4y - 5 = 0$ $r_3: -4x + 6y + 7 = 0$
State which of the following lines are parallel and which are perpendicular to each other:
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$r_1: 2x - 3y + 1 = 0$
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$r_2: 6x + 4y - 5 = 0$
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$r_3: -4x + 6y + 7 = 0$
Lines with a Parameter
If a line depends on a parameter $k$, the condition for parallelism or perpendicularity turns into an equation for $k$. If an exact angle is given, set the formula for $\tan\alpha$ equal to the required value.
Calculate the values of the parameter $k$ so that the line $kx + y - 2 = 0$ forms an angle of $45^\circ$ with the line with 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 with equation $x - 2y + 1 = 0$.
Two aeroplanes follow straight-line paths 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 $k$…
Two aeroplanes follow straight-line paths 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 along straight 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 along straight rails. The first follows the line $r: 3x + 4y - 12 = 0$ and the second the line $s: ax + 8y - 5 = 0$.
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Determine $a$ so that the rails are parallel.
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Find the equation of a third transverse rail that is perpendicular to both and crosses rail $r$ exactly at its y-intercept.
Applications: Altitudes, Reflections, and Structures
Perpendicularity shows up whenever something must "drop at a right angle": the altitude of a triangle, a ray reflected in a mirror, a crossbeam, or the projection of a point onto a line.
Reflection of a ray in a mirror
Law of reflection: the incident ray and the reflected ray form the same angle with the normal, which is the line perpendicular to the mirror at the point where the ray hits it. If the mirror is horizontal, the normal is vertical and the reflected ray keeps the same inclination but the sign of its slope changes.
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 side streets $x + y - 2 = 0$ and $2x - y - 4 = 0$. Obtain the general equation of the new avenue.
In the plan for a metallic structure, a main support beam rests on the line passing through nodes $P(-2, 4)$ and $Q(4, 1)$. A second beam needs to be welded…
In the plan for a metallic structure, a main support beam rests on the line passing through 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 explicit equation of the line defining this second beam.
Given a triangle with vertices $A(1, 2)$, $B(5, 3)$ and $C(4, 6)$, find the general equation of the altitude corresponding to 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 vertex $A$ and calculate the angle this altitude forms with the side $AB$.
Find the coordinates of the orthogonal projection of the origin of coordinates $(0,0)$ onto the line with equation $2x - 5y + 10 = 0$.
Find the coordinates of the orthogonal projection of the origin of coordinates $(0,0)$ onto the line with equation $2x - 5y + 10 = 0$.
A laser beam travels along the line $y = -2x + 5$ and reflects off a flat mirror situated 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 flat mirror situated 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.
Distance and Properties of the Rhombus
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.
The diagonals of a rhombus lie on 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 its side length is…
The diagonals of a rhombus lie on 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 its side length is $\sqrt{20}$, find the coordinates of the other three vertices.
More Complete Problems
Three useful ideas for the hard exercises:
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Angle bisectors. They are the points that are equidistant from the two lines: $\dfrac{|A_1x+B_1y+C_1|}{\sqrt{A_1^2+B_1^2}}=\dfrac{|A_2x+B_2y+C_2|}{\sqrt{A_2^2+B_2^2}}$. Removing the absolute value gives two lines, perpendicular to each other.
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Symmetric line with respect to another: take the point where both lines intersect and the image of another point on $r$ (drop the perpendicular to the axis of symmetry and extend it the same distance).
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Family of lines with a parameter: group by powers of $k$, $(\dots)+k(\dots)=0$, and the fixed point solves the system formed by both parentheses.
One vertex of an equilateral triangle is the origin $(0,0)$ and the opposite side lies on the line $x + y - 4 = 0$. Determine the general equations of the lines…
One vertex of an equilateral triangle is the origin $(0,0)$ and the opposite side lies on 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.
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 gradients, that both bisectors are perpendicular to each other.
A solar panel is aligned with the line $r: \sqrt{3}x - y + 10 = 0$. The sun's rays strike 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 strike 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 about its intersection point with the $OY$-axis so that the rays hit completely perpendicular to its surface, what will be the new general equation of the line defining the panel?
Back to the beginning: two rails are parallel if their slopes are equal, perpendicular if the product is $-1$, and in any other case the angle comes from $\tan\alpha$.