Equations of a line in the plane
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 $\frac{x-2}{2} = \frac{y+1}{-2}$.
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-gradient 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 $4x + 3y - 10 = 0$.