Equations of a line in the plane
Picture a Luas line on a map of Dublin: on a short stretch it runs almost dead straight. If you know the coordinates of two stops, can you write one equation that describes every point on that stretch of track? You will answer this by the end of the page.
The same line can be written in six different ways, and textbooks and exams jump from one to another without warning. These are not six lines: it is the same line in six different "disguises". If you understand where each form comes from, converting between them is mechanical.
The line as a point and a direction: vector equation
Write the vector equation and parametric equations of the line passing through the point $A(3, -2)$ with direction vector $\vec{v} = (-1, 4)$.
Write the vector equation and parametric equations of the line passing through the point $A(3, -2)$ with direction vector $\vec{v} = (-1, 4)$.
Splitting into $x$ and $y$: parametric equations
A surveillance drone moves in a straight line at a constant speed from point $A(10, 20)$ to point $B(50, 80)$, with the coordinates measured in metres. Write the…
A surveillance drone moves in a straight line at a constant speed from point $A(10, 20)$ to point $B(50, 80)$, with the coordinates measured in metres. Write the parametric equations of its trajectory and state its coordinates when it has travelled exactly half the distance.
An underground pipe follows the path given by the parametric equations $x = -5 + 2t$, $y = 12 - 3t$ (in km). Determine the exact points at which the pipe crosses the…
An underground pipe follows the path given by the parametric equations $x = -5 + 2t$, $y = 12 - 3t$ (in km). Determine the exact points at which the pipe crosses the coordinate axes.
Eliminating the parameter: continuous equation
Check whether the points $M(4, -3)$ and $N(1, 2)$ belong to the line with continuous equation $\frac{x-2}{2} = \frac{y+1}{-2}$.
Check whether the points $M(4, -3)$ and $N(1, 2)$ belong to the line with continuous equation $\frac{x-2}{2} = \frac{y+1}{-2}$.
Given the line in parametric equations $x = 2 - 3t$, $y = 1 + 5t$, express it in continuous form and in general form.
Given the line in parametric equations $x = 2 - 3t$, $y = 1 + 5t$, express it in continuous form and in general form.
In a Mathematics exam, Cian must write the continuous equation of the line that passes through the point $A(-3, 5)$ and has direction vector $\vec{v} = (2, -4)$. On his…
In a Mathematics exam, Cian must write the continuous equation of the line that passes through the point $A(-3, 5)$ and has direction vector $\vec{v} = (2, -4)$. On his sheet, he writes:
$$\frac{x - 3}{2} = \frac{y + 5}{-4}$$When grading it, his teacher points out that the answer contains a widespread conceptual error.
- a) Identify precisely the two sign mistakes Cian made in the numerators of the expression.
- b) Explain the origin of this error by deriving the continuous equation step by step from the parametric equations: first write $x = a_1 + t\times v_1$ and $y = a_2 + t\times v_2$ with the given data and solve for the parameter $t$ in each one to show why the signs must be the opposite of what Cian wrote.
- c) Write the correct continuous equation and, from it, obtain the general equation of the line by simplifying the integer coefficients as much as possible.
Clearing denominators: general equation
Write the implicit equations of the horizontal line passing through the point $H(-7, 4)$ and the vertical line passing through the point $V(3, -9)$.
Write the implicit equations of the horizontal line passing through the point $H(-7, 4)$ and the vertical line passing through the point $V(3, -9)$.
Given the implicit equation $5x + 7y - 11 = 0$, state a direction vector, a normal vector, and a point through which the line passes.
Given the implicit equation $5x + 7y - 11 = 0$, state a direction vector, a normal vector, and a point through which the line passes.
Find the general equation of the line passing through the point $A(-4, 6)$ and parallel to the line connecting the points $B(1, 1)$ and $C(3, -5)$.
Find the general equation of the line passing through the point $A(-4, 6)$ and parallel to the line connecting the points $B(1, 1)$ and $C(3, -5)$.
The line as a function: explicit and point-slope equations
Determine the gradient and the y-intercept of the line whose general equation is $4x - 2y + 8 = 0$.
Determine the gradient and the y-intercept of the line whose general equation is $4x - 2y + 8 = 0$.
Write the point-gradient equation of the line passing through the points $P(-1, 5)$ and $Q(2, -1)$.
Write the point-gradient equation of the line passing through the points $P(-1, 5)$ and $Q(2, -1)$.
The production cost of an electronic component follows a linear model. Manufacturing $100$ units costs $450$ €, and manufacturing $250$ units costs $825$ €. Determine…
The production cost of an electronic component follows a linear model. Manufacturing $100$ units costs $450$ €, and manufacturing $250$ units costs $825$ €. Determine the explicit equation relating the cost $y$ to the number of units $x$, and calculate the fixed production cost (the cost when $0$ units are manufactured).
The six faces of the same line
Express the line $y = -\frac{1}{4}x + 2$ in vector form and find three distinct points that belong to it, where at least one has a negative x-coordinate.
Express the line $y = -\frac{1}{4}x + 2$ in vector form and find three distinct points that belong to it, where at least one has a negative x-coordinate.
Determine the values of the parameters $a$ and $b$ so that the line given in parametric form $x = a + 2t$, $y = 3 - bt$ represents the same line as the general equation…
Determine the values of the parameters $a$ and $b$ so that the line given in parametric form $x = a + 2t$, $y = 3 - bt$ represents the same line as the general equation $4x + 3y - 10 = 0$.
A laser beam starts from the point $A(2, 5)$, hits a flat mirror situated along the x-axis, reflects, and passes through the point $B(8, 3)$. Knowing that the angle of…
A laser beam starts from the point $A(2, 5)$, hits a flat mirror situated along the x-axis, reflects, and passes through the point $B(8, 3)$. Knowing that the angle of incidence is equal to the angle of reflection, find the equations of the lines containing the incident beam and the reflected beam, and the exact bounce point on the mirror.
Back to the start: with the coordinates of two stops, find the slope $m=\frac{y_2-y_1}{x_2-x_1}$, then write $y-y_1=m(x-x_1)$. That one equation describes every point on the straight stretch of track.