Question

Find the symmetric equations and the parametric equations of a line, and/or the equation of the plane satisfying the following given conditions. Line through (0,-2,4) and (3,-2,-1)

Short answer

Direction vector: (3, 0, -5). Parametric: \ x = 3t, \ y = -2, \ z = 4 - 5t. Symmetric: \[ \frac{x}{3} = \frac{z-4}{-5} \]

Step-by-step solution

- Find the direction vector

To find the direction vector \(\text{\textbf{d}}\), subtract the coordinates of the initial point (0, -2, 4) from the coordinates of the terminal point (3, -2, -1). This gives us: \[ \text{\textbf{d}} = (3 - 0, -2 - (-2), -1 - 4) = (3, 0, -5) \]

- Form the parametric equations

Using the point (0, -2, 4) and the direction vector \((3, 0, -5)\), write the parametric equations of the line as follows: \[ x = 0 + 3t, \ y = -2 + 0t, \ z = 4 - 5t \]

- Form the symmetric equations

To form the symmetric equations, eliminate the parameter \(t\). From the parametric equations we get: \[ t = \frac{x}{3}, \ t = \frac{z-4}{-5} \] Since \ y = -2 \ is constant, the symmetric equations are: \[ \frac{x}{3} = \frac{z-4}{-5} \]

Key concepts

Direction Vector

In 3D geometry, a direction vector is crucial in defining the orientation of a line. To find the direction vector \(\text{\textbf{d}}\) of a line passing through two points, we subtract the coordinates of the starting point from the ending point. For example, given points (0,-2,4) and (3,-2,-1), we find:
\[ \text{\textbf{d}} = (3 - 0, -2 - (-2), -1 - 4) = (3, 0, -5) \]
This vector \( (3, 0, -5) \) indicates the direction of the line in 3D space.

Parametric Equations

Parametric equations translate the geometric notion of a line into algebraic form. They express each coordinate (x, y, z) in terms of a parameter, usually denoted by \( t \).
For the line through (0,-2,4) with direction vector (3,0,-5), the parametric equations are:
\[ x = 0 + 3t \] \[ y = -2 + 0t \] \[ z = 4 - 5t \]
Here, \( t \) represents any real number, allowing you to find different points on the line by plugging in various values of \( t \). Notice, \( y = -2 \) remains constant because the direction vector's y-component is 0.

Symmetric Equations

Symmetric equations eliminate the parameter \( t \) from parametric equations to express a single relationship among x, y, and z.
From the parametric equations:
\[ t = \frac{x}{3} \] \[ t = \frac{z-4}{-5} \]
Equate these to form the symmetric equation:
\[ \frac{x}{3} = \frac{z-4}{-5} \]
Since \( y = -2 \) is constant, the final symmetric equation does not include \( y \). Symmetric equations compactly describe the line's orientation and location in 3D space.

Coordinate Geometry

Coordinate geometry, or analytic geometry, allows geometric problems to be solved using algebraic methods.
Points in 3D space are represented as coordinates \( (x, y, z) \). Lines can be described by equations derived from coordinates and vectors. Here's a quick guide:

• Start with two points to find the direction vector by subtracting coordinates.
• Use a point and direction vector to write parametric equations.
• Convert parametric equations to symmetric equations for a concise line description.

Understanding these steps greatly simplifies many problems in 3D geometry, making it easier to visualize and solve spatial problems.