Problem 9 - Advanced Engineering Mathematics

Question

In Problems, find parametric equations for the line through the given points. $$ (1,0,0),(3,-2,-7) $$

Step-by-Step Solution

Answer

Parametric equations are \(x = 1 + 2t\), \(y = 0 - 2t\), \(z = 0 - 7t\).

Step 1: Find the direction vector

To find the parametric equations of a line through two points, we begin by determining the direction vector. If the points are \((x_1, y_1, z_1) = (1, 0, 0)\) and \((x_2, y_2, z_2) = (3, -2, -7)\), the direction vector \(\mathbf{d}\) is found by subtracting the coordinates: \(\mathbf{d} = (x_2-x_1, y_2-y_1, z_2-z_1) = (3-1, -2-0, -7-0) = (2, -2, -7)\). This vector \((2, -2, -7)\) will guide the line's direction.

Step 2: Set up parametric equations

Now that we have the direction vector \((2, -2, -7)\), we set up parametric equations using one of the points, for instance, \((1, 0, 0)\). The parametric equations will be formulated as: \(x = 1 + 2t\), \(y = 0 - 2t\), \(z = 0 - 7t\), where \(t\) is the parameter.

Key Concepts

Direction Vector

When we want to describe a line in three-dimensional space, one essential component we need is the direction vector. Think of the direction vector as an arrow pointing in the line's direction. It gives us the orientation of the line in space.

A direction vector can be found by subtracting the coordinates of two points on the line. For example, if you have two points

• \((x_1, y_1, z_1) = (1, 0, 0)\)
• \((x_2, y_2, z_2) = (3, -2, -7)\)

the direction vector \(\mathbf{d}\) is determined by the differences:

• \(x_2 - x_1 = 3 - 1 = 2\)
• \(y_2 - y_1 = -2 - 0 = -2\)
• \(z_2 - z_1 = -7 - 0 = -7\)

Thus, the direction vector for our example is \((2, -2, -7)\).

This vector tells us that for every unit increase in \(t\), the line moves 2 units in the \(x\)-direction, \(-2\) units in the \(y\)-direction, and \(-7\) units in the \(z\)-direction. This conceptual understanding will help in writing parametric equations for the line.

Line Through Two Points

Creating a line through two points requires using both the points' locations and the direction vector. Imagine you have a starting point on the line and a direction in which the line stretches. This is the core idea of constructing a line through two points in space.

In our case, the points are

• \((1, 0, 0)\)
• \((3, -2, -7)\)

The direction vector calculated from these points is

• \((2, -2, -7)\)

Armed with this direction vector, we can plot a line starting at one of our points, let's say

• \((1, 0, 0)\)

Then, using a parameter \(t\), we can create equations that express how each coordinate of the line changes.

The line can be written in the parametric form:

• \(x = 1 + 2t\)
• \(y = 0 - 2t\)
• \(z = 0 - 7t\)

As \(t\) changes, these equations describe how points on the line move away from our initial point in the direction indicated by our direction vector.

Vector Subtraction

Vector subtraction is a straightforward but essential mathematical operation when working with lines through points. It's the action of finding a vector that points from one point to another.

To perform vector subtraction:

• Take two points \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\)
• Subtract their corresponding coordinates: \(x_2 - x_1\), \(y_2 - y_1\), \(z_2 - z_1\)

In the given problem:

• The first point is \((1, 0, 0)\)
• The second point is \((3, -2, -7)\)

The direction vector is calculated as:

• \((3 - 1, -2 - 0, -7 - 0) = (2, -2, -7)\)

This vector subtraction yields the direction in which the line proceeds, or simply put, it tells us the step-by-step movement along the line, from the first point to the second.

Vector subtraction is a fundamental process that often underpins more complex operations in vector mathematics and helps establish a concrete understanding of orientation in space.