How to find the parametric equation of a line?
Understand the Problem
The question is asking for the method to derive parametric equations for a line, which involves understanding how to express lines in terms of a parameter, typically using endpoint coordinates and a direction vector.
Answer
The parametric equations are: $$ x(t) = x_1 + (x_2 - x_1) t $$ $$ y(t) = y_1 + (y_2 - y_1) t $$ for \( 0 \leq t \leq 1 \).
Answer for screen readers
The parametric equations for the line defined by points ( A(x_1, y_1) ) and ( B(x_2, y_2) ) are:
$$ x(t) = x_1 + (x_2 - x_1) t $$
$$ y(t) = y_1 + (y_2 - y_1) t $$
for ( 0 \leq t \leq 1 ).
Steps to Solve
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Identify the endpoints of the line Begin by determining the coordinates of the two points that define the line. Suppose point ( A ) has coordinates ( (x_1, y_1) ) and point ( B ) has coordinates ( (x_2, y_2) ).
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Calculate the direction vector Find the direction vector ( \vec{d} ) of the line by subtracting the coordinates of point ( A ) from point ( B ): $$ \vec{d} = (x_2 - x_1, y_2 - y_1) $$
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Set up the parametric equations The parametric equations for the line can be expressed using a parameter ( t ). The equations are: $$ x(t) = x_1 + (x_2 - x_1) t $$ $$ y(t) = y_1 + (y_2 - y_1) t $$ Here, ( t ) typically varies from 0 to 1.
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State the parameter range Define the range of ( t ). For the complete segment between points ( A ) and ( B ), set ( t ) to go from 0 to 1: $$ 0 \leq t \leq 1 $$
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Complete the parametric representation The complete parametric representation of the line is given as: $$ \begin{cases} x(t) = x_1 + (x_2 - x_1) t \ y(t) = y_1 + (y_2 - y_1) t \end{cases} $$
The parametric equations for the line defined by points ( A(x_1, y_1) ) and ( B(x_2, y_2) ) are:
$$ x(t) = x_1 + (x_2 - x_1) t $$
$$ y(t) = y_1 + (y_2 - y_1) t $$
for ( 0 \leq t \leq 1 ).
More Information
Parametric equations allow us to represent curves and lines in a way that treats each coordinate as a function of a parameter. This is particularly useful in physics and computer graphics where motion along a path can be described with time as the parameter.
Tips
- Forgetting to specify the range of the parameter ( t ). Always ensure to specify if it covers the exclusive or inclusive range for correct interpretation.
- Mixing up the endpoint coordinates when calculating the direction vector. It's important to consistently use the order of points to avoid negative direction vectors.
