91Ó°ÊÓ

To understand parametric equations of a line in space, it's essential to grasp the concept of the direction vector. A direction vector is a vector that gives the direction of the line. This vector can be obtained by subtracting the coordinates of two points that lie on the line.
In this specific exercise, we used the points \((1, 5, 2)\) and \((5, 0, -1)\). By calculating the vector \((x_2-x_1, y_2-y_1, z_2-z_1)\), we find the direction vector of the line: \((4, -5, -3)\).
This direction vector is significant because it determines the slope and tilt of the line in the three-dimensional space. Once the direction vector is known, you can use it, along with a point on the line, to write the parametric equations.

A line in space is characterized by its path through an infinite set of points. Unlike a line on a plane, a line in space has a direction given by its direction vector, and it can extend in any orientation—up, down, side-to-side, or tilted.
In the realm of parametric equations, a line in space is described using one point on the line and a direction vector. The parametric form allows us to systematically express and explore this infinite path. By using a parameter, typically denoted by \( t \), we can represent every point on the line.
The parametric equations, derived from both a point and a direction vector, provide a compact representation of the entire line. In this exercise, the line through the points \((1, 5, 2)\) and \((5, 0, -1)\) is expressed using these equations: \[ \begin{align*} x = 1 + 4t, \ y = 5 - 5t, \ z = 2 - 3t \end{align*} \]

Vector subtraction is a crucial operation used to find the direction vector of a line. When you subtract one position vector from another, you obtain a new vector that points from one point to the other.
This concept is illustrated in the exercise where we needed to find a direction vector from two given points. By using the coordinate subtraction formula \((x_2-x_1, y_2-y_1, z_2-z_1)\), we subtract the coordinates of the initial point \((1, 5, 2)\) from the coordinates of the terminal point \((5, 0, -1)\). This gives us the direction vector \((4, -5, -3)\).
This mathematical subtraction not only provides a means to direction but also a step into writing parametric equations, as shown in the solution.

The coordinates of a point are fundamental in geometry and vector analysis. They describe the exact location of a point in space and consist of an ordered set—typically (x, y, z) in three-dimensional space.
In this exercise, two sets of coordinates \((1, 5, 2)\) and \((5, 0, -1)\) are given. These coordinates are pivotal in determining both the line and its direction in space.
Each coordinate signifies a specific dimension: 'x' for horizontal position, 'y' for vertical position, and 'z' for depth. By using these coordinates, we were able to calculate the direction vector and proceed to derive the parametric equations.