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 \((4,1,0)\)

Step-by-Step Solution

Answer

The parametric equations are: \[x = 2 + 2t, \quad y = t, \quad z = -3 + 3t.\]

Step 1: Identify Direction Vector

To find the parametric equation of a line in space, we first must determine the direction vector. This vector can be found by subtracting the coordinates of the first point from the second point:\[\vec{d} = (4 - 2, 1 - 0, 0 + 3) = (2, 1, 3)\]

Step 2: Define Parametric Equations

With the direction vector and one point, we can write the parametric equations. Generally, the parametric equations of a line that passes through a point \(x_0, y_0, z_0\) with direction vector \(a, b, c\) are \[x = x_0 + at\y = y_0 + bt\z = z_0 + ct\]

Step 3: Write Parametric Equations for This Line

Using the point \(2, 0, -3\) and direction vector \(2, 1, 3\), we substitute to find the equations:\[x = 2 + 2t\y = 0 + 1t\z = -3 + 3t\]

Step 4: Complete Parametric Equations

Thus, the parametric equations describing the line that goes through the given points are:\[x = 2 + 2t\y = t\z = -3 + 3t\]

Key Concepts

Direction Vector

The concept of a direction vector is crucial when working with parametric equations in three-dimensional geometry. A direction vector essentially describes the direction and the "slope" of a line. When given two points, like \( (2, 0, -3) \) and \( (4, 1, 0) \), we determine the direction by subtracting corresponding coordinates of these points.

• Start with the x-coordinates: \( 4 - 2 = 2 \)
• Next, the y-coordinates: \( 1 - 0 = 1 \)
• Finally, the z-coordinates: \( 0 - (-3) = 3 \)

The direction vector \( \vec{d} \) is then \( (2, 1, 3) \), meaning the line moves 2 units in the x-direction, 1 unit in the y-direction, and 3 units in the z-direction for each step along the line. This vector is a key component in forming the parametric equations of a line.

3D Coordinate Geometry

Three-dimensional coordinate geometry involves analyzing shapes, lines, and points in a space defined by three axes: x, y, and z. This makes it possible to describe and visualize objects and their relations in a space with depth, unlike the traditional 2D plane. In 3D geometry, understanding vectors and how they function can greatly simplify problems. Instead of viewing a line as a series of points, we think of it as a direction governed by its direction vector.

• The point coordinates, such as \( (2, 0, -3) \), give us the starting point in the space.
• The direction vector \( (2, 1, 3) \) then propels us in a particular direction within this space.

We use this comprehensive understanding to derive equations and solve problems related to lines, planes, and more, all organized within the three-dimensional framework.

Line in Space

Describing a line in the three-dimensional space involves parametric equations, which rely on both a point on the line and a direction vector. These aspects ensure that every point on the line can be represented. The parametric equations for a line involve assigning each coordinate – x, y, and z – as a function of a parameter, often denoted as \( t \). For example, given \( (2, 0, -3) \) as a point and \( (2, 1, 3) \) as a direction vector, we form equations:\[ x = 2 + 2t \]\[ y = 0 + 1t \ (or \ y = t) \]\[ z = -3 + 3t \]This means that, by altering \( t \), you move along the line uniquely mapped out in the 3D space. Parametric equations have a distinctive advantage in that they offer a straightforward way to traverse the entirety of any line in space, allowing for flexible and practical use in various applications, from graphics to engineering design.