Problem 11 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Find the Direction Vector

To find the parametric equations for the line, first determine the direction vector \(\vec{d}\) of the line by subtracting the coordinates of point \(P\) from point \(Q\). Compute \(Q - P = (0 - 1, 2 - 1, 2 - 0) = (-1, 1, 2)\). This vector \((-1, 1, 2)\) represents the direction of the line.

Step 2: Write the Parametric Equations

Using point \(P\) and the direction vector \(\vec{d} = (-1, 1, 2)\), write the parametric equations for each coordinate. The equations are given by: - \(x = 1 - t\) - \(y = 1 + t\) - \(z = 0 + 2t\), where \(t\) is the parameter.

Key Concepts

Direction Vector

In the world of 3D geometry, understanding direction vectors is key to describing lines comprehensively. A direction vector essentially points in the direction in which the line extends. It doesn't matter where on the line you choose your start or end point; the direction will help describe the whole line.
To find a direction vector between two points, such as those in your exercise, you subtract the coordinates of the initial point from the final point. Consider point \( P(1, 1, 0) \) and point \( Q(0, 2, 2) \). The direction vector is calculated as:
\[ Q - P = (0-1, 2-1, 2-0) = (-1, 1, 2) \]

• The first component \(-1\) tells us the line moves 1 unit in the opposite direction on the x-axis per step.
• The second component \(1\) indicates it moves 1 unit in the same direction on the y-axis per step.
• The third component \(2\) denotes movement 2 units in the same direction on the z-axis per step.

This vector \((-1, 1, 2)\) serves to define the path of the line in 3D space.

Coordinate Geometry

Coordinate geometry uses numbers to represent geometric figures and the relationships among them. In the realm of parametric equations for 3D lines, we express each of the x, y, and z coordinates as separate functions of a parameter, often represented as \(t\). This method is powerful because it gives a comprehensive view of the line in three dimensions.

Using our points in the exercise, we apply the direction vector to the initial point’s coordinates to get the following parametric equations:

• \( x = 1 - t \)
• \( y = 1 + t \)
• \( z = 0 + 2t \)

Here, \(t\) is a parameter that can be any real number. As \(t\) changes, the point \( (x, y, z) \) traces the entire line. Each parametric equation shows how that particular coordinate changes in relation to the parameter.

3D Lines

3D lines translate the idea of straight pathways from the 2-dimensional plane into three-dimensional space. Visualizing them can seem tricky, but by breaking them down into coordinates, as we do with parametric equations, it becomes easier to understand their path and orientation.

For lines in 3D space, we use a known point through which the line passes, along with a direction vector that captures the line’s slope or orientation. In our exercise, the point \( P(1, 1, 0) \) acts as the starting point. From there, the direction of \((-1, 1, 2)\) guides the line's path.

• The use of three separate equations allows us to represent complex spatial relationships simply.
• Parametric forms are particularly useful in computer graphics and calculus where mapping motion and defining precise paths are crucial.

Understanding 3D lines through parametric equations not only aids in math problems but also has practical applications in fields such as engineering and computer science.