Equation of a line
To describe a street on a city map, it is enough to give a point it passes through and the direction it goes in. In mathematics it is the same: a point and a direction vector determine a single line, and that line can be written in several ways depending on what you want to highlight.
A point and a direction vector
Given the general equation of the line $r: 3x - 5y + 7 = 0$, a student quickly claims that its direction vector is $\vec{v} = (3, -5)$. Identify the mistake made when…
Given the general equation of the line $r: 3x - 5y + 7 = 0$, a student quickly claims that its direction vector is $\vec{v} = (3, -5)$. Identify the mistake made when extracting information from the general equation and determine a correct direction vector and normal vector for this line.
Vector and parametric equations
Write the vector equation and parametric equations of the line that passes through point $A(2, -5)$ and has direction vector $\vec{v} = (3, 4)$.
Write the vector equation and parametric equations of the line that passes through point $A(2, -5)$ and has direction vector $\vec{v} = (3, 4)$.
Determine the coordinates of two different points lying on the line whose parametric equations are: $x = 1 - t$ $y = 2 + 3t$
Determine the coordinates of two different points lying on the line whose parametric equations are:
$x = 1 - t$
$y = 2 + 3t$
Continuous equation and general equation
If you solve for $t$ in both parametric equations and equate them, you obtain the continuous equation; if you clear the denominators and collect terms, you reach the general equation:
$$\text{continuous:}\ \frac{x-a_1}{v_1} = \frac{y-a_2}{v_2} \qquad\longrightarrow\qquad \text{general:}\ Ax+By+C=0$$Express in general form the line given by the following symmetric equation: $\frac{x-3}{2} = \frac{y+1}{-5}$
Express in general form the line given by the following symmetric equation:
$\frac{x-3}{2} = \frac{y+1}{-5}$
Identify the coordinates of a point and the components of a direction vector of the line given by the equation: $\frac{x+4}{-3} = \frac{y-2}{7}$
Identify the coordinates of a point and the components of a direction vector of the line given by the equation:
$\frac{x+4}{-3} = \frac{y-2}{7}$
Explicit equation
The most familiar form, $y = mx+n$, comes from solving for $y$ in the general equation: $m$ (the gradient, in the next section) is $-A/B$ and $n$ is $-C/B$.
Write the explicit equation of the line that passes through point $B(4, 1)$ and has gradient $m = -\frac{1}{2}$.
Write the explicit equation of the line that passes through point $B(4, 1)$ and has gradient $m = -\frac{1}{2}$.
Given the line with equation $3x - 4y + 12 = 0$, find the value of its gradient and its y-intercept.
Given the line with equation $3x - 4y + 12 = 0$, find the value of its gradient and its y-intercept.
Membership and intersection of lines
A point lies on a line if the equation is satisfied when you substitute its coordinates. And the point where two lines cross is the one that satisfies both equations at once: you solve the system.
Verify analytically whether points $C(3, 2)$ and $D(-1, 4)$ lie on the line with equation $2x + y - 8 = 0$.
Verify analytically whether points $C(3, 2)$ and $D(-1, 4)$ lie on the line with equation $2x + y - 8 = 0$.
Two observation satellites follow straight-line trajectories described by equations $r: 3x + y - 7 = 0$ and $s: x - 2y + 4 = 0$. Calculate the exact coordinates of the…
Two observation satellites follow straight-line trajectories described by equations $r: 3x + y - 7 = 0$ and $s: x - 2y + 4 = 0$. Calculate the exact coordinates of the point where their routes intersect.