Equation of a Line Passing Through Given Points in 3D Space

When dealing with lines in three-dimensional space, the process of determining the equation of a line passing through given points requires a bit more complexity than working in two dimensions. In this article, we’ll explore how to find the equation of a line passing through the points 1, 2, 3 and 3, 4, 5 using both parametric and vector forms.

Understanding the Problem

We are given two points in three-dimensional space: (P_1(1, 2, 3)) and (P_2(3, 4, 5)). The goal is to find the equation of the line that passes through these points. This is a common problem in various fields, including physics, engineering, and computer graphics.

Method 1: Parametric Form of the Line Equation

The parametric form of the line equation in 3D can be useful in many applications. It allows us to express the coordinates of any point on the line as a function of a parameter (t).

Necessary Calculation: The direction vector dd can be found by subtracting the coordinates of (P_1) from (P_2):

d P2-P1 3 - 1 4 - 2 5 - 3 d 3 - 1 4 - 2 5 - 3

d 2, 2, 2d 2, 2, 2

Using the point (P_1(1, 2, 3)) and the direction vector (d(2, 2, 2)), the parametric equations of the line can be written as:

Parametric Equations

The vector form of the line equation can also be written as:

Vector Form

Conclusion

In summary, the line passing through the points 1, 2, 3 and 3, 4, 5 can be represented by the parametric equations:

Parametric Equations

Or in vector form:

Vector Form

Additional Insights

While the provided solution may initially seem to use different methods, all valid as long as they adhere to the principles of linear algebra. The different approaches—parametric form, vector form, and the point-slope form—offer different perspectives on the same problem, and can be useful in various contexts.

Understanding these methods is not only important for solving problems, but also for developing a deeper understanding of the geometric relationships between points and lines in space.