Problem 3 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the parametric equations of the line through the given pair of points. $$(4,2,3),(6,2,-1)$$

Step-by-Step Solution

Answer

\( x = 4 + 2t, \, y = 2, \, z = 3 - 4t \)

Step 1: Identify given points

We have two points given: \( (4, 2, 3) \) and \( (6, 2, -1) \). These points will help us find the direction vector of the line.

Step 2: Find the direction vector

The direction vector is calculated by subtracting the coordinates of the first point from the second. \[ (6 - 4, 2 - 2, -1 - 3) = (2, 0, -4) \]

Step 3: Write the parametric equations

Using the point \((4, 2, 3)\) as a reference point and the direction vector \((2, 0, -4)\), the parametric equations of the line are:\[ x = 4 + 2t \]\[ y = 2 + 0t \] (or simply \( y = 2 \))\[ z = 3 - 4t \]

Step 4: Conclusion

The parametric equations of the line passing through the points \((4, 2, 3)\) and \((6, 2, -1)\) are:\[ x = 4 + 2t, \, y = 2, \, z = 3 - 4t \].

Key Concepts

Direction Vector

In 3D coordinate geometry, a direction vector of a line is crucial for understanding the nature of lines in space. It essentially indicates the line's orientation. By deciphering how one point goes from one position to another, we define this vector. If we have two points, say \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\), the direction vector \(\textbf{d}\) can be found by subtracting the coordinates of the first point from those of the second:

• Subtracting x-coordinates: \((x_2 - x_1)\)
• Subtracting y-coordinates: \((y_2 - y_1)\)
• Subtracting z-coordinates: \((z_2 - z_1)\)

This results in the vector \(\textbf{d} = (x_2 - x_1, y_2 - y_1, z_2 - z_1)\). This vector shows the change across each axis between the two points and is fundamental in expressing the parametric form of the line.

3D Coordinate Geometry

3D coordinate geometry expands on 2D geometry by adding an additional dimension, the z-axis, which creates a three-dimensional space. This allows for:

• Positions to be identified in terms of \((x, y, z)\) coordinates
• The representation of objects with volume
• Exploration of geometric concepts using planes, lines, and surfaces

Understanding 3D geometry is essential for visualizing lines in this space. Each point is represented using three coordinates. Lines and vectors need additional descriptions because they can travel in more directions. Parametric equations, which utilize a direction vector along with a point, are therefore used to describe lines. These equations tell us everything about the path a line takes through space.

Line Through Points

The concept of a line through points in 3D space involves defining a path that connects two particular points. These points provide the necessary information to calculate both the direction and representation of the line. When given two points, like in our example:

• Point A: \((4, 2, 3)\)
• Point B: \((6, 2, -1)\)

We can determine the line's direction using the direction vector. Once we have the direction vector, \((2, 0, -4)\) in this case, we can write the line's parametric equations. These equations use a parameter, usually \(t\), to describe every point on the line:

• For the x-coordinate: \(x = 4 + 2t\)
• For the y-coordinate: \(y = 2\)
• For the z-coordinate: \(z = 3 - 4t\)

These parametric equations fully describe the line through the given points, allowing for clear visual and analytical understanding of the line's path.