Equation of Straight Line in Parametric Form

The equation of a straight line can be expressed in various forms, such as slope-intercept form, point-slope form, two-point form, and general form. Another important form is the parametric form, which is particularly useful in three-dimensional geometry and when dealing with vectors.

Understanding Parametric Equations

Parametric equations define a group of quantities as functions of one or more independent variables called parameters. In the context of straight lines, parametric equations allow us to express the coordinates of any point on the line as functions of a single parameter, usually denoted by ( t ).

Parametric Form of a Straight Line

Consider a line ( L ) in two-dimensional space. Let ( \mathbf{a} = (x_1, y_1) ) be a fixed point on ( L ), and let ( \mathbf{v} = (v_x, v_y) ) be a direction vector parallel to ( L ). The parametric equations of the line passing through ( \mathbf{a} ) and parallel to ( \mathbf{v} ) are given by:

[ \begin{align*} x &= x_1 + v_x t \ y &= y_1 + v_y t \end{align*} ]

where ( t ) is the parameter.

Table of Differences and Important Points

Aspect	Parametric Form	Other Forms (e.g., Slope-Intercept)
Representation	Uses a parameter ( t )	Does not use a parameter
Flexibility	Can easily represent vertical lines	Vertical lines are problematic
Application	Useful in 3D geometry and with vectors	Common in 2D geometry
Conversion to/from	Can be converted to/from other forms	Can be converted to parametric form
Visualization	Represents points as a function of ( t )	Represents y as a function of x

Formulas

For a line in three-dimensional space, the parametric equations extend to include the ( z )-coordinate:

[ \begin{align*} x &= x_1 + v_x t \ y &= y_1 + v_y t \ z &= z_1 + v_z t \end{align*} ]

where ( \mathbf{a} = (x_1, y_1, z_1) ) is a point on the line, ( \mathbf{v} = (v_x, v_y, v_z) ) is a direction vector, and ( t ) is the parameter.

Examples

Example 1: Two-Dimensional Space

Given a line passing through the point ( (2, 3) ) with a direction vector ( (1, 2) ), find the parametric equations of the line.

Solution:

Let ( \mathbf{a} = (2, 3) ) and ( \mathbf{v} = (1, 2) ). The parametric equations are:

[ \begin{align*} x &= 2 + 1 \cdot t = 2 + t \ y &= 3 + 2 \cdot t = 3 + 2t \end{align*} ]

Example 2: Three-Dimensional Space

Find the parametric equations of a line passing through ( (1, -2, 4) ) and parallel to the vector ( (3, 0, -1) ).

Solution:

Let ( \mathbf{a} = (1, -2, 4) ) and ( \mathbf{v} = (3, 0, -1) ). The parametric equations are:

[ \begin{align*} x &= 1 + 3t \ y &= -2 + 0t = -2 \ z &= 4 - 1t = 4 - t \end{align*} ]

Converting to Other Forms

To convert parametric equations to other forms, you can eliminate the parameter ( t ). For example, if you have the parametric equations ( x = 2 + t ) and ( y = 3 + 2t ), you can solve the first equation for ( t ) to get ( t = x - 2 ) and substitute it into the second equation to obtain ( y = 3 + 2(x - 2) ), which simplifies to ( y = 2x - 1 ), the slope-intercept form.

Conclusion

The parametric form of a straight line is a powerful tool in geometry, especially when dealing with lines in higher dimensions or when incorporating vector concepts. It provides a way to describe the position of a point on a line as a function of a single parameter, offering flexibility and ease of use in various mathematical and physical contexts.