How to find the parametric equation of a line?
Understand the Problem
The question is asking how to derive the parametric equations that represent a line in space. This typically involves identifying a point on the line and a direction vector, and then expressing the coordinates of points on the line using a parameter.
Answer
The parametric equations of a line in space are $$ \begin{cases} x = x_0 + at \\ y = y_0 + bt \\ z = z_0 + ct \end{cases} $$
Answer for screen readers
The parametric equations of a line in space are $$ \begin{cases} x = x_0 + at \ y = y_0 + bt \ z = z_0 + ct \end{cases} $$ where $(x_0, y_0, z_0)$ is a point on the line and $(a, b, c)$ is the direction vector.
Steps to Solve
- Identify a Point on the Line
To derive the parametric equations of a line in space, start by identifying a point on the line, say $P_0(x_0, y_0, z_0)$.
- Determine the Direction Vector
Next, determine the direction vector of the line, denoted as $\mathbf{d} = (a, b, c)$, where $a$, $b$, and $c$ are the components of the vector. This vector indicates the direction in which the line extends.
- Set Up the Parametric Equations
Using the point and the direction vector, you can formulate the parametric equations for the line. The equations are: $$ x = x_0 + at $$ $$ y = y_0 + bt $$ $$ z = z_0 + ct $$
Here, $t$ is a parameter that varies over all real numbers.
- Combine the Equations
You can express the equations together as: $$ \begin{cases} x = x_0 + at \ y = y_0 + bt \ z = z_0 + ct \end{cases} $$
These equations give the coordinates of any point on the line as $t$ changes.
The parametric equations of a line in space are $$ \begin{cases} x = x_0 + at \ y = y_0 + bt \ z = z_0 + ct \end{cases} $$ where $(x_0, y_0, z_0)$ is a point on the line and $(a, b, c)$ is the direction vector.
More Information
Parametric equations are a fundamental concept in vector calculus and are used to represent lines and curves in a space. Each parameterization allows you to visualize the line and compute intersections with other geometric objects.
Tips
- Forgetting to include the point on the line: Always ensure to identify a specific point before setting up the equations.
- Confusing direction vector components: It's important to correctly identify the components of the direction vector; they should correspond to the increments in each coordinate direction.
