Exercises: Parallel and perpendicular lines

Write down the gradient of a line that is parallel and the gradient of a line that is perpendicular to the line with equation $y = 4x - 7$.

Given the lines $r: y = -2x + 5$ and $s: y = \frac{1}{2}x - 3$, state with reasons whether they are parallel, perpendicular or intersecting but not perpendicular.

Find the explicit equation of the line that is parallel to $y = -3x + 2$ and passes through the point $P(0, 5)$.

A direction vector of the line $r$ is $\vec{v} = (3, -1)$. Write down a direction vector for a line that is parallel to $r$ and another for a line that is perpendicular to $r$.

Determine the point-gradient equation of the line perpendicular to $y = \frac{2}{5}x - 1$ that passes through the point $A(4, 2)$.

Check analytically if the lines given in general form $r: 3x - 6y + 2 = 0$ and $s: x - 2y - 5 = 0$ are parallel.

Write down the general equation of the line that is parallel to the x-axis and passes through the point $Q(-3, 8)$.

Calculate the gradient of the line passing through the points $A(2, -1)$ and $B(5, 8)$, and deduce from it what the gradient of any line perpendicular to it would be.

On a city map, Main Street follows the path of the line $2x + 5y - 10 = 0$. Market Street is perpendicular to it and passes through a square located at the coordinates $P(-2, 3)$. Find the general equation of Market Street.

Calculate the value of the parameter $k$ so that the line $kx - 3y + 6 = 0$ is parallel to the line joining the points $A(1, 2)$ and $B(4, 11)$.

A laser beam travels along the line $y = 3x - 2$. At the exact point where the laser crosses the y-axis, a mirror is placed perpendicular to the path of the beam. Determine the explicit equation of the line representing the mirror.

The adjacent sides of a rectangular plot lie on the lines $r: 2x - 4y + 7 = 0$ and $s: 4x + 2y - 3 = 0$. Show analytically that the sides form a right angle. Then, find the equation of a line parallel to $r$ that passes through the origin.

Two aeroplanes fly at the same altitude. The route of the first aeroplane is given by the parametric equations $x = 1 + 5t$, $y = -2 - 3t$. The route of the second aeroplane is parallel to the first and passes through the control tower $T(0, 0)$. Write down the parametric equations of the route of the second aeroplane.

Determine the value of the parameter $m$ so that the lines $r: mx + 4y - 1 = 0$ and $s: 2x - 5y + 8 = 0$ are perpendicular.

A main irrigation pipe follows the line with continuous equation $\frac{x-2}{4} = \frac{y+1}{-3}$. A secondary pipe needs to be installed perpendicular to the main one, connecting to a sprinkler located at $S(-1, 5)$. Find the continuous equation of this secondary pipe.

Find the explicit equation of the perpendicular bisector of the segment with endpoints $A(-2, 3)$ and $B(4, 1)$. (Remember that the perpendicular bisector is the line perpendicular to the segment that passes through its midpoint).

The roof design of a sports hall includes two crossing steel beams. Beam 1 rests on the line $ax + 3y - 5 = 0$ and beam 2 on the line $2x - by + 4 = 0$. Calculate the exact values of $a$ and $b$ knowing that the beams are perpendicular and that beam 1 passes through the point $P(1, -2)$.

Prove analytically that the quadrilateral whose vertices are $A(1, 1)$, $B(5, 3)$, $C(4, 5)$ and $D(0, 3)$ is a rectangle. To do this, check the gradients of its sides and confirm that the adjacent angles are right angles.

A ship sails in a straight line following the path $r: 3x + 4y - 24 = 0$. A lighthouse is located at the point $F(7, 7)$. Calculate the shortest distance from the lighthouse to the ship's path. (Hint: find the line perpendicular to $r$ that passes through $F$, find the point of intersection and calculate the distance between both points).

Given the lines $r: \frac{x+1}{2} = \frac{y-3}{k}$ and $s: 3x - 5y + 2 = 0$, determine the value of $k$ so that they are parallel. Then, calculate the value of $k$ so that they are perpendicular.

Calculate the exact coordinates of the orthocenter (point of intersection of the three altitudes) of the triangle with vertices $A(-2, 1)$, $B(4, 3)$ and $C(2, -5)$.

A solar panel is aligned with the line passing through the points $P(1, 4)$ and $Q(5, 2)$. To maximize efficiency, the sun's rays must strike the panel perpendicularly. Write down the parametric equations of the path of a sunbeam that hits exactly at the midpoint of the panel.

Find the exact coordinates of the orthogonal projection of the point $P(3, -2)$ onto the line $r: x - 2y + 5 = 0$. (Note: the orthogonal projection is the point of intersection of the line $r$ with the line perpendicular to it that passes through $P$).

Two robot vacuum cleaners move around an open-plan room. The first follows the path given by the equation $y = \frac{1}{2}x + 2$. The second moves along a line perpendicular to the first, such that they would collide at the exact point of intersection of the first path with the y-axis. Find the explicit equation of the second robot's path and calculate the area of the triangle formed by the paths of both robots and the x-axis.