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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblema 12.1.95a
Capitolo 12, Problema 12.1.95a

Intersecting lines Consider the following pairs of lines. Determine whether the lines are parallel or intersecting. If the lines intersect, then determine the point of intersection.


a. x = 1 + s, y = 2s and x = 1 + 2t, y = 3t

Guida verificata passo dopo passo
1
Identify the parametric equations of the two lines. The first line is given by \(x = 1 + s\) and \(y = 2s\), and the second line is given by \(x = 1 + 2t\) and \(y = 3t\), where \(s\) and \(t\) are parameters.
To check if the lines intersect, set the \(x\) and \(y\) coordinates equal to each other because at the point of intersection both lines share the same coordinates. So, solve the system: \(1 + s = 1 + 2t\) and \(2s = 3t\).
From the first equation, simplify to find a relationship between \(s\) and \(t\): \(s = 2t\).
Substitute \(s = 2t\) into the second equation \(2s = 3t\) to get \(2(2t) = 3t\), which simplifies to \(4t = 3t\).
Analyze the resulting equation to determine if there is a solution for \(t\). If a solution exists, use it to find \(s\) and then substitute back into the parametric equations to find the coordinates of the intersection point. If no solution exists, conclude that the lines are parallel and do not intersect.

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Parametric Equations of Lines

Parametric equations express the coordinates of points on a line as functions of a parameter, typically denoted as t or s. Each parameter value corresponds to a unique point on the line, allowing a clear representation of lines in the plane or space.
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Parameterizing Equations

Determining Parallelism of Lines

Two lines are parallel if their direction vectors are scalar multiples of each other. By comparing the coefficients of the parameters in the parametric equations, one can check if the lines have the same direction, indicating parallelism.
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Finding the Point of Intersection

If lines are not parallel, their point of intersection can be found by equating their parametric expressions and solving the resulting system of equations for the parameters. Substituting these values back gives the coordinates of the intersection point.
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