Line Through Points in 3D Space
Find the line through the points (5, -2, 3) and (7, 4, 1)
To find the equation of a line through two points, the key idea is to first determine the direction of the line. This is done by calculating a vector that points from one point to the other. Conceptually, you're figuring out how much the coordinates change as you move from the first point to the second. Once you have this direction vector, it tells you the "slope" of the line in three dimensions.
With the direction vector in hand, you can then describe the line using parametric equations. These equations express each coordinate—x, y, and z—as a function of a parameter, showing how they change along the line. To write these equations, you need any point on the line (one of the given points works) and the direction vector. From there, you can also transform the parametric form into symmetric equations, which is another way to express the line by relating the changes in each coordinate to one another without the parameter.