Problem 59 - Calculus for Biology and Medicine

Question

In Problems \(59-62\), find the parametric equation of the line in \(x-\) \(y-z\) space that goes through the given points. \((5,4,-1)\) and \((2,0,3)\)

Step-by-Step Solution

Answer

The parametric equations are: \(x = 5 - 3t\), \(y = 4 - 4t\), and \(z = -1 + 4t\).

Step 1: Understand the Parametric Equation for a Line

In three-dimensional space, a line passing through a point \((x_0, y_0, z_0)\) with direction vector \((a, b, c)\) can be described by parametric equations:\[x = x_0 + at, \, y = y_0 + bt, \, z = z_0 + ct\]where \(t\) is a parameter.

Step 2: Determine the Direction Vector

To find the direction vector \((a, b, c)\), subtract the coordinates of the first point from the second point. For points \((5, 4, -1)\) and \((2, 0, 3)\), the direction vector is:\[(2 - 5, \, 0 - 4, \, 3 - (-1)) = (-3, \, -4, \, 4)\]

Step 3: Write the Parametric Equations

Using the point \((5, 4, -1)\) and the direction vector \((-3, -4, 4)\), the parametric equations are:- \(x = 5 - 3t\)- \(y = 4 - 4t\)- \(z = -1 + 4t\)

Key Concepts

Direction Vector

The direction vector is a fundamental concept when dealing with lines in geometry, especially in 3D space. It essentially tells us the direction in which a line is oriented. If you're tasked with finding a line that passes through two points in space, the direction vector is calculated by taking the difference between the coordinates of these two points.
For example, if we have two points \(A(5, 4, -1)\) and \(B(2, 0, 3)\), we would find the direction vector by subtracting the coordinates of \(A\) from those of \(B\).

• Direction along x: \(2 - 5 = -3\)
• Direction along y: \(0 - 4 = -4\)
• Direction along z: \(3 - (-1) = 4\)

Thus, the direction vector is \((-3, -4, 4)\). This vector is crucial for forming the parametric equations of a line, acting like a compass that guides which way the line stretches in space.

3D Space

Understanding 3D space is vital for solving problems involving parametric equations of lines. In mathematical terms, 3D space is depicted with three axes, usually labeled x, y, and z. These axes are perpendicular to each other, helping to visualize the position and orientation of points and lines in this expansive environment.
The points in 3D space are given by ordered triples (x, y, z), representing positions along the x-axis, y-axis, and z-axis respectively. Unlike 2D, where a point only needs an x and a y coordinate, 3D incorporates depth with the z-coordinate. This allows us to model more realistic spaces and movements occurring in real life.
For students, getting familiar with drawing and imagining 3D space is often the first step in tackling parametric equations and understanding the vast and exciting world of 3D geometry.

Line Equations

Line equations in 3D space can appear intimidating, but breaking them down into parametric equations often simplifies the problem. In the context of 3D space, a line can be represented using a point it passes through and a direction vector that defines its orientation.
The parametric form of a line equation is expressed as:

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

Here, \((x_0, y_0, z_0)\) is a point on the line, and \((a, b, c)\) is the direction vector. The variable \(t\) is a parameter, representing different points on the line as \(t\) varies. For instance, if \((x_0, y_0, z_0) = (5, 4, -1)\) and \((a, b, c) = (-3, -4, 4)\), the line equations become:

• \(x = 5 - 3t\)
• \(y = 4 - 4t\)
• \(z = -1 + 4t\)

These equations are simple and elegant, defining a line that stretches infinitely in both directions through the given point while following the path set by the direction vector.