How to find parametric equations of a line?
Understand the Problem
The question is asking for the method to derive the parametric equations that represent a straight line in mathematics. Parametric equations involve expressing the coordinates of points on the line as functions of a parameter.
Answer
The parametric equations of the line segment are: $$ \begin{align*} x(t) &= (1 - t) x_1 + t x_2 \\ y(t) &= (1 - t) y_1 + t y_2 \end{align*} $$ for \( t \in [0, 1] \).
Answer for screen readers
The parametric equations that represent a straight line between two points ( (x_1, y_1) ) and ( (x_2, y_2) ) are: $$ \begin{align*} x(t) &= (1 - t) x_1 + t x_2 \ y(t) &= (1 - t) y_1 + t y_2 \end{align*} $$ for ( t ) in the interval ( [0, 1] ).
Steps to Solve
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Identify the endpoints of the line segment
Determine two points through which the line passes. Let’s denote these points as $(x_1, y_1)$ and $(x_2, y_2)$. -
Define the parameter
Choose a parameter to represent the line. A common choice is ( t ), where ( t ) varies between 0 and 1. When ( t = 0 ), the parametric equations will give you the point ( (x_1, y_1) ). When ( t = 1 ), the equations will yield ( (x_2, y_2) ). -
Create the parametric equations
Develop the parametric equations for the x and y coordinates based on the endpoints. The equations can be expressed as: $$ x(t) = (1 - t) x_1 + t x_2 $$ $$ y(t) = (1 - t) y_1 + t y_2 $$ -
Express the equations clearly
Write the final parametric equations clearly, indicating that these equations represent the coordinates of points along the line segment between the two endpoints as ( t ) varies from 0 to 1: $$ \begin{align*} x(t) &= (1 - t) x_1 + t x_2 \ y(t) &= (1 - t) y_1 + t y_2 \end{align*} $$
The parametric equations that represent a straight line between two points ( (x_1, y_1) ) and ( (x_2, y_2) ) are: $$ \begin{align*} x(t) &= (1 - t) x_1 + t x_2 \ y(t) &= (1 - t) y_1 + t y_2 \end{align*} $$ for ( t ) in the interval ( [0, 1] ).
More Information
These parametric equations allow us to describe the line continuously as a function of the parameter ( t ), making them very useful in various fields such as physics and computer graphics for animating movements along a path.
Tips
- Confusing the roles of ( x_1, y_1, x_2, y_2 ) when substituting values into the parametric equations. Ensure that ( (x_1, y_1) ) is the starting point and ( (x_2, y_2) ) is the endpoint.
- Forgetting to specify the range of ( t ). Always mention that ( t ) should vary from 0 to 1 for it to trace the segment correctly.
