Problem 56 - Calculus for Biology and Medicine

Question

Find the parametric equation of the line in \(x-y-z\) space that goes through the indicated point in the direction of the indicated vector. $$(2,0,4),\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right]$$

Step-by-Step Solution

Answer

The parametric equation is \(x = 2 + t\), \(y = 2t\), \(z = 4 + 3t\).

Step 1: Understand the Problem

We need to find the parametric equation of a line that passes through a given point \((x_0, y_0, z_0) = (2, 0, 4)\) and moves in the direction of a given vector \(\mathbf{d} = \begin{bmatrix} 1 \ 2 \ 3 \end{bmatrix}\). The line equation should be expressed in terms of a parameter \(t\).

Step 2: Write the General Parametric Equation of a Line

The parametric equation of a line that passes through a point \((x_0, y_0, z_0)\) with a direction vector \(\mathbf{d} = \begin{bmatrix} a \ b \ c \end{bmatrix}\) is given by: \[ x = x_0 + at, \ y = y_0 + bt, \ z = z_0 + ct \] where \(t\) represents the parameter.

Step 3: Substitute the Given Point and Direction Vector

Substitute the given point \((x_0, y_0, z_0) = (2, 0, 4)\) and direction vector \(\mathbf{d} = \begin{bmatrix} 1 \ 2 \ 3 \end{bmatrix}\) into the general parametric equation:\[ x = 2 + 1t, \ y = 0 + 2t, \ z = 4 + 3t \]

Step 4: Simplify the Equations

The parametric equations simplify to:\[ x = 2 + t, \ y = 2t, \ z = 4 + 3t \]These represent the parametric equation of the line in the \(x-y-z\) space.

Key Concepts

Vector Directions

Understanding vector directions is key in the realm of 3D geometry. Vectors are entities that possess both magnitude and direction. They are often used to convey how far and in what direction a certain point should move. When dealing with lines in space, the direction vector is a critical component. It indicates the direction in which the line extends. However, it does not fix the length you need to travel along the line. Instead, it shows the direction you "shoot" off from the given point.

In the case of directional vectors, such as \( \begin{bmatrix} 1 \ 2 \ 3 \end{bmatrix} \), each component influences its respective coordinate axis: the x-component represents how fast the line progresses along the x-axis, the y-component for the y-axis, and so on. This multiplication by parameter \( t \) scales the length of the vector but not its direction. Thus ensuring the line can be infinitely traced in either direction.

3D Coordinate Geometry

3D coordinate geometry goes beyond the flat, 2D space that consists of x and y axes to include a third dimension: the z-axis. This creates a coordinate system that can comprehend volumes and spatial situations. Each point in this 3D space is represented by a triple of coordinates \((x, y, z)\). These coordinates describe positions within this space, establishing locations in a cube-like format rather than a flat plane.

This context allows representation of points, lines, and shapes spatially. For example, lines aren't just plotted across a plane but can stretch infinitely in the deepest layer of the human understanding. Behavior of these lines in space is expressed via equations including all three axes. Thus, 3D coordinate geometry serves as the foundation for various applications, from engineering designs to video game graphics.

Lines in Space

Lines in space can be incredibly versatile and are not limited to flat surfaces. They can tilt, rise, and dive, which upgrades our ability to navigate and visualize multi-dimensional spaces. When expressing such a line analytically, a parametric equation is often deployed. This expression resembles:

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

In this set, \((x_0, y_0, z_0)\) defines a point the line passes through, while the vector components \(a, b, c\) articulate its direction in space.

The parameter \(t\) acts as a slider. By choosing different values for \(t\), you can precisely identify any point along the line. Whether \(t\) is positive, negative, or zero, it effectively represents a location on the line, allowing both positive and negative directions from the initial point. This methodology completely defines the line's trajectory in an understandable manner in 3D space.