Problem 62 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The parametric equation is \( (x, y, z) = (1 + 2t, 2t, 4 - 4t) \).

Step 1: Identify two points

The problem provides two points through which the line passes: \( (1, 0, 4) \) and \( (3, 2, 0) \). Use these points to determine the direction vector of the line.

Step 2: Determine direction vector

The direction vector \( \vec{d} \) from point \( (1,0,4) \) to point \( (3,2,0) \) can be found by subtracting the coordinates of the first point from the second:\[ \vec{d} = (3-1, 2-0, 0-4) = (2, 2, -4) \].

Step 3: Write the parametric equations

To write the parametric equations of the line, use the first point and the direction vector. If \( t \) is a parameter, the parametric equations are:\[x = 1 + 2t,y = 0 + 2t,z = 4 - 4t\]

Key Concepts

Direction Vector

In geometry, the direction vector plays a crucial role when representing lines in space. For any line, a direction vector provides information about the path it takes. To find it, you need two points that the line passes through.
In our exercise, the two given points are

• Point A: (1, 0, 4)
• Point B: (3, 2, 0)

The direction vector, often denoted as \( \vec{d} \), can be found by subtracting the coordinates of point A from point B. This gives us \( (3-1, 2-0, 0-4) \), resulting in \( \vec{d} = (2, 2, -4) \).
This vector tells us how to move from one point to the next along the line and therefore influences the line's orientation in space.

Line in Space

A line in space, unlike a line in a plane, exists in three-dimensional (3D) space. It traverses the dimensions represented by the axes \( x \), \( y \), and \( z \). Understanding a line's existence in 3D is crucial for solving equations involving spatial objects.
The line's path through a 3D space can be determined by its direction vector and a given point on the line. Using our example, starting from point A:

• Point A: (1, 0, 4)

combined with the direction vector \( (2, 2, -4) \), the line can be fully described in its path through the 3D space.
This combination ensures that no matter how far you project the line, it will maintain its direction defined by the vector.

Parametric Form

The parametric form of an equation is a powerful tool for describing lines, especially in 3-dimensional space. This form utilizes a parameter, often denoted as \( t \), which allows the calculation of the exact position on a line at any given point.
To transform a line into its parametric form, you start with a known point on the line and incorporate its direction vector. Using our given point A:

• (1, 0, 4)

and direction vector \( (2, 2, -4) \), we describe the line in parametric form as:

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

Each equation describes how the \( x \), \( y \), and \( z \) coordinates change as \( t \) varies, thus mapping every point along the line.