Problem 60 - Calculus for Biology and Medicine

Question

Find the parametric equation of the line in \(x-y-z\) space that goes through the given points. \((2,0,-3)\) and \((4,1,1)\)

Step-by-Step Solution

Answer

The parametric equations are \(x = 2 + 2t\), \(y = t\), \(z = -3 + 4t\).

Step 1: Find the Direction Vector

Identify the direction vector of the line by subtracting the first point from the second point. With point A as \((2, 0, -3)\) and point B as \((4, 1, 1)\), calculate the direction vector \(\vec{d}\): \(\vec{d} = (4-2, 1-0, 1-(-3)) = (2, 1, 4)\).

Step 2: Write the Parametric Equations

Using the direction vector \((2, 1, 4)\) and a point on the line, say \((2, 0, -3)\), write the parametric equations. The parametric equations of the line are given by: \(x = 2 + 2t\), \(y = 0 + 1t\), and \(z = -3 + 4t\), where \(t\) is a parameter.

Key Concepts

Direction Vector

When we talk about a direction vector, we are referring to a vector that gives us the orientation or direction of a line. In our specific exercise, to find the direction vector, we need to calculate the difference between two given points. Imagine starting from point A and moving to point B in 3D space. The direction vector is a package of all the changes you make in each direction:

• In the x-direction, move from 2 to 4, resulting in a change of \(4 - 2 = 2\).
• In the y-direction, go from 0 to 1, making a change of \(1 - 0 = 1\).
• Finally, in the z-direction, move from -3 to 1, which changes by \(1 - (-3) = 4\).

Putting these changes together, the direction vector for our line is \(\vec{d} = (2, 1, 4)\). This vector tells you how far to move in the x, y, and z directions to go along the line. Think of it as your compass and pace notes all in one!

3D Space

3D space, short for three-dimensional space, is a place that adds depth to the usual flat, two-dimensional world. Just like how we see our surroundings, it consists of three axes:

• The x-axis, which typically measures horizontal position.
• The y-axis, commonly showing vertical position.
• The z-axis, which adds the depth, making it the third dimension.

Every point in 3D space is given by a triplet of numbers \((x, y, z)\). These numbers tell you how to reach the point from a reference point, usually called the origin \((0, 0, 0)\), by moving along the x, y, and z directions.
In our exercise, points like \(2, 0, -3\) and \(4, 1, 1\) represent specific locations in this 3D world. Understanding how these points relate spatially involves visualizing them in this three-dimensional coordinate system.

Line Equation

The line equation in 3D provides a mathematical description of a line by using points and direction. A popular form to represent it is the parametric equation. It uses a parameter \(t\) to describe points along the line. This parameter acts like a slider, adjusting it slides you along the line. For our exercise, given a point \((2, 0, -3)\) and direction vector \((2, 1, 4)\), the parametric equations for the line are:

• \(x = 2 + 2t\)
• \(y = 0 + 1t\)
• \(z = -3 + 4t\)

These equations tell you where to find points on the line for any value of \(t\). As \(t\) changes, it scales the direction vector, effectively moving us along the line and pinpointing every possible location on it. This concept is essential for understanding how to map a line through 3D space.