Problem 10 - Algebra and Trigonometry

Question

Find parametric equations for the line that passes through the points \(P\) and \(Q .\) $$ P(2,-1,-2), \quad Q(0,1,-3) $$

Step-by-Step Solution

Answer

The parametric equations are: \(x = 2 - 2t\), \(y = -1 + 2t\), \(z = -2 - t\).

Step 1: Identify the direction vector

To find the parametric equations for the line, we first need the direction vector of the line. This vector is obtained by subtracting the coordinates of point \(P\) from point \(Q\). The vector \( \mathbf{v} \) from \(P\) to \(Q\) is given by: \[ \mathbf{v} = Q - P = (0-2, 1+1, -3+2) = (-2, 2, -1) \]

Step 2: Write the parametric equations

With the direction vector \( \mathbf{v} = (-2, 2, -1) \) and a point either \(P\) or \(Q\) on the line, we can write the parametric equations. Using \(P(2, -1, -2)\) as the known point, the parametric equations for the line are: \[ x = 2 - 2t \] \[ y = -1 + 2t \] \[ z = -2 - t \] Here, \(t\) is the parameter that varies along the line.

Key Concepts

Direction Vector

The direction vector is crucial for describing the path and orientation of a line in space. It essentially acts as a "compass" that indicates the line's direction.
To determine it, you take the difference between the coordinates of two given points on the line. This tells us which way the line is heading. In our exercise, the points provided were \( P(2, -1, -2) \) and \( Q(0, 1, -3) \).
We calculate the direction vector \( \mathbf{v} \) by subtracting the coordinates of \( P \) from \( Q \):

• \( v_x = Q_x - P_x = 0 - 2 = -2 \)
• \( v_y = Q_y - P_y = 1 - (-1) = 2 \)
• \( v_z = Q_z - P_z = -3 - (-2) = -1 \)

Thus, the direction vector is \( \mathbf{v} = (-2, 2, -1) \). This vector forms the backbone of our parametric equations.

Line Equations

Line equations in three-dimensional space describe how points move along a line. They require a fixed point on the line and a direction vector.
The parametric form is a popular way to express these line equations, using equations in terms of a parameter \( t \). The line's behavior is captured neatly by this parameter.
For our example, we use point \( P(2, -1, -2) \) and the direction vector \( \mathbf{v} = (-2, 2, -1) \). The parametric equations are:

• \( x = 2 - 2t \)
• \( y = -1 + 2t \)
• \( z = -2 - t \)

Here, \( t \) governs the movement along the line. Each value of \( t \) gives us a new point on the line. This collection of equations captures the entirety of the line’s direction perfectly.

3D Geometry

Three-dimensional geometry involves understanding shapes, lines, and planes in a space that we can explore in a full range of directions. A line in 3D space isn't just an extended path; it's a specific trajectory connecting different points using axes \( x, y, \) and \( z \).
Unlike 2D lines, 3D lines require more information due to their additional degrees of freedom. Generally, they’re defined by a point they pass through and a direction vector. The direction vector gives the essential inclination or steepness in multi-dimensional space.
Visualizing these ideas involves imagining the coordinate system and the line darting through it, perfectly aligning with its defining direction vector. Understanding how these elements interact is integral to mastering 3D geometry.

Parametric Representation

Parametric representation is a method of defining mathematical entities using parameters. In the context of lines, it means representing the coordinates \( x, y, \) and \( z \) using equations dependent on parameter \( t \).
This approach elegantly encapsulates how a line extends infinitely in both directions. Each point on the line corresponds to a specific value of \( t \), making it simple to calculate new positions.
For example, in our exercise, the parametric equations for the line are:

• \( x = 2 - 2t \)
• \( y = -1 + 2t \)
• \( z = -2 - t \)

By adjusting \( t \), you smoothly trace the line through space. This flexibility and clarity make parametric representation an invaluable tool in geometry and beyond.