When studying linear algebra and calculus, one of the fundamental concepts is to find the equation of a line. There are different ways to represent the equation of a line, but one of the most common methods is to use vector and parametric equations. In this article, we will explore how to find a vector equation and parametric equations for a line using the parameter t.
What is a Vector Equation?
A vector equation is a way to represent a line in three-dimensional space using vectors. A vector is a mathematical quantity that has both magnitude and direction. In the case of a line, we can use two vectors to describe its position and direction. To find the vector equation of a line, we need to know a point on the line and its direction vector.
Let's say we have a line L that passes through the point (1,2,3) and has a direction vector d = a i + b j + c k. The vector equation of the line L is:
r(t) = a + td
where r(t) is the position vector of any point on the line, t is the parameter that varies along the line, a is the initial point on the line, and d is the direction vector of the line.
What are Parametric Equations?
Parametric equations are another way to represent the equation of a line. Instead of using vectors, parametric equations use a set of equations that describe the position of a point on the line as a function of a parameter t. The parameter t is a variable that represents the distance along the line.
Let's say we have a line L that passes through the point (1,2,3) and has a direction vector d = a i + b j + c k. The parametric equations of the line L are:
x = 1 + at
y = 2 + bt
z = 3 + ct
where x, y, and z are the coordinates of any point on the line, t is the parameter that varies along the line, and a, b, and c are the direction ratios of the line.
How to Find the Vector Equation and Parametric Equations of a Line?
To find the vector equation and parametric equations of a line, we need to follow these steps:
- Find a point on the line.
- Find the direction vector of the line.
- Use the point and direction vector to find the vector equation of the line.
- Use the point and direction ratios to find the parametric equations of the line.
Let's illustrate this process with an example.
Example: Find the vector equation and parametric equations of the line passing through the points (1,2,3) and (4,5,6).
- Find a point on the line.
- Find the direction vector of the line.
- Use the point and direction vector to find the vector equation of the line.
- Use the point and direction ratios to find the parametric equations of the line.
We can choose either of the two given points to be a point on the line. Let's choose (1,2,3).
The direction vector of the line is the vector that connects the two given points. We can find it by subtracting the coordinates of the two points.
d = p2 - p1 = <4,5,6> - <1,2,3> = <3,3,3>
Using the point (1,2,3) and the direction vector <3,3,3>, we can write the vector equation of the line as:
r(t) = <1,2,3> + t<3,3,3>
The direction ratios of the line are a = 3/b = 3/c = 3. Using the point (1,2,3) and the direction ratios, we can write the parametric equations of the line as:
x = 1 + 3t
y = 2 + 3t
z = 3 + 3t
These are the vector equation and parametric equations of the line passing through the points (1,2,3) and (4,5,6).
Conclusion
Vector and parametric equations are useful ways to represent the equation of a line in three-dimensional space. To find the vector equation and parametric equations of a line, we need to know a point on the line and its direction vector or direction ratios. By following the steps outlined in this article, we can easily find the vector equation and parametric equations of any line in three-dimensional space.
