Problem 3 - University Calculus: Early Transcendentals

Question

Find parametric equations for the lines. The line through \(P(-2,0,3)\) and \(Q(3,5,-2)\)

Step-by-Step Solution

Answer

The parametric equations are: \( x = -2 + 5t \), \( y = 5t \), \( z = 3 - 5t \).

Step 1: Understand the Problem

We need to find the parametric equations of a line passing through two given points: \( P(-2, 0, 3) \) and \( Q(3, 5, -2) \). Each point gives us coordinates in 3D space.

Step 2: Calculate the Direction Vector

The direction vector \( \vec{d} \) of the line is found by subtracting the coordinates of point \( P \) from \( Q \). So, \( \vec{d} = (3-(-2), 5-0, -2-3) = (5, 5, -5) \).

Step 3: Write the Parametric Equations

The parametric equations for a line using a point \( A(x_0, y_0, z_0) \) and direction vector \( \vec{d} \) can be expressed as: \( x = x_0 + at \), \( y = y_0 + bt \), \( z = z_0 + ct \). For this problem: \( x = -2 + 5t \), \( y = 0 + 5t \), \( z = 3 - 5t \), where \( t \) is the parameter.

Key Concepts

Direction Vector

The concept of a direction vector is central when dealing with parametric equations of lines in 3D space. Imagine vectors as arrows in space, showing direction and magnitude. To find a direction vector for a line, you need two points through which the line passes. In our problem, we use points \( P(-2, 0, 3) \) and \( Q(3, 5, -2) \). These points provide us with the necessary coordinates to find the direction.
To calculate the direction vector \( \vec{d} \), subtract the coordinates of point \( P \) from those of point \( Q \). This yields the vector:

• In the x-direction: \( 3 - (-2) = 5 \)
• In the y-direction: \( 5 - 0 = 5 \)
• In the z-direction: \( -2 - 3 = -5 \)

Thus, the direction vector \( \vec{d} \) is \( (5, 5, -5) \). It tells us how fast and in what direction the line moves through space in each axis.

3D Space Coordinates

3D space and its coordinates can seem intricate, but breaking it down proves helpful. In three-dimensional space, any point is expressed as a triplet: \( (x, y, z) \). These coordinates define a point's position as:

• \( x \): how far the point is along the x-axis
• \( y \): how far the point is along the y-axis
• \( z \): how far the point is along the z-axis

Considering our line between \( P(-2, 0, 3) \) and \( Q(3, 5, -2) \), these coordinates place these points precisely in 3D space. The starting coordinate, \( P \), serves as a reference point for constructing the parametric equations.

Line Equations

Line equations provide a way to represent lines algebraically. In the context of 3D geometry, parametric equations are particularly useful. Every line has these equations, expressed in terms of a parameter, often \( t \). Given a point \( A(x_0, y_0, z_0) \) and a direction vector \( \vec{d}(a, b, c) \), the parametric equations are:

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

These equations allow us to calculate the exact position on the line for any given \( t \). For our line, using \( P(-2, 0, 3) \) as the initial point and \( \vec{d} = (5, 5, -5) \) as the direction vector, these equations become:

• \( x = -2 + 5t \)
• \( y = 0 + 5t \)
• \( z = 3 - 5t \)

The parameter \( t \) slides the point along the line, expanding or contracting the vector from the initial point.

Linear Algebra

Linear algebra forms the backbone of many mathematical concepts, including those involving lines in space. It deals primarily with vectors, matrices, and linear transformations, fundamental tools for understanding and manipulating lines and planes.
When finding parametric equations, linear algebra assists in vividly describing the line's direction and position within space. By employing vectors and coordinates, it simplifies complex spatial problems into manageable algebraic equations.
In our scenario, using linear algebra principles, we derived the direction vector \( \vec{d} \) and incorporated point \( P(-2, 0, 3) \) to establish our line's parametric equations. Such applications are pivotal in physics, computer graphics, and engineering, providing solutions to real-world spatial challenges.