Angle between two lines. Parallel and perpendicular lines
When two lines cross, they form a specific angle. That angle can be calculated without a protractor and without drawing anything: the gradients or direction vectors of the two lines are enough. It is the tool industrial robots use to check that two rails are perpendicular, or that an architect uses so that two walls form the exact angle in the plan.
The angle between two lines
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 $s: 3x + y - 1 = 0$.
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 $y = -3x + 4$.
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
When a line depends on a parameter, the conditions for parallel and perpendicular lines become an equation that determines that parameter, and the condition for an exact angle is expressed by applying the tangent formula.
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 appears whenever something has to "fall at a right angle": the altitude of a triangle onto its side, a ray reflected in a mirror, a crossbeam that reinforces a structure.
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.
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 $AB$.
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.