Problem 4 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the parametric equations of the line through the given pair of points. $$(5,-3,-3),(5,4,2)$$

Step-by-Step Solution

Answer

The parametric equations are: \(x = 5\), \(y = -3 + 7t\), \(z = -3 + 5t\).

Step 1: Determine a Direction Vector

To find the parametric equations of a line through two points, first calculate the direction vector. Given the points \((5, -3, -3)\) and \((5, 4, 2)\), subtract the coordinates of the first point from the corresponding coordinates of the second: \[\mathbf{d} = (5 - 5, 4 - (-3), 2 - (-3)) = (0, 7, 5).\] This vector \((0, 7, 5)\) serves as the direction vector of the line.

Step 2: Write the Parametric Equations

Utilize the direction vector found in Step 1 and the original point \((5, -3, -3)\) to write the parametric equations of the line. The general form for parametric equations using a point \((x_0, y_0, z_0)\) and a direction vector \((a, b, c)\) is:\[x = x_0 + at,\]\[y = y_0 + bt,\]\[z = z_0 + ct.\]Plugging in the given point and direction vector:\[x = 5 + 0t,\]\[y = -3 + 7t,\]\[z = -3 + 5t.\]

Step 3: Simplify the Parametric Equations

Simplify the equations from Step 2 where possible. Since the x-component has a direction of zero, it remains constant:\[x = 5,\]\[y = -3 + 7t,\]\[z = -3 + 5t.\]These are the parametric equations of the line through the given points.

Key Concepts

Direction Vector

A direction vector is essential in defining a line in 3D space, as it indicates the direction in which the line extends. When you are given two points, like (5, -3, -3) and (5, 4, 2), finding the direction vector involves calculating the difference between these points. This process is known as vector subtraction.

• For each coordinate, subtract the corresponding coordinate of the first point from the second.
• The direction vector \( \mathbf{d} = (0, 7, 5) \) in this case.

This vector represents not only the direction of the line but also its slope in 3D space. This is the first step to determining the parametric equation of the line.

Parametric Line Equation

The parametric line equation is crucial as it provides a way to express the line using parameters. This approach is particularly useful in 3D geometry because a 3D line is not easily expressed with just one equation.

• Start with a point on the line (called the point vector), such as (5, -3, -3).
• Combine it with the direction vector \( (0, 7, 5) \) from the line's equation.

In the parametric form, a line is expressed as:

• \( x = x_0 + at \)
• \( y = y_0 + bt \)
• \( z = z_0 + ct \)

Where \( t \) is the parameter that can assume any real number value. The parametric equations tell us how to reach each point on the line.

3D Coordinate Geometry

3D coordinate geometry deals with points, lines, and shapes in three-dimensional space. Unlike 2D, you have an additional dimension, making it challenging but very interesting.

• It involves using the x, y, and z axes to define the position of points.
• Lines in 3D are defined by direction vectors and a point or parametric equations.

Understanding these equations and operations allows us to navigate and model 3D space, providing insights into how objects interact in a three-dimensional world. For example, knowing how to form a line in 3D helps in visualizing paths and understanding spatial relationships.

Vector Subtraction

Vector subtraction is the method used to determine the direction vector between two points. In our context, if you need the direction vector between the points (5, -3, -3) and (5, 4, 2):

• Subtract the x, y, and z components of the first point from the corresponding elements of the second point.
• Resulting in the direction vector \( \mathbf{d} = (0, 7, 5) \).

This process highlights how vector subtraction helps in identifying directions and differences in space. This basic operation in vector mathematics forms the foundation for more complex concepts like calculating parametric line equations and understanding 3D geometry.