Problem 55 - 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. $$(1,-1,2),\left[\begin{array}{r}1 \\ -2 \\ 1\end{array}\right]$$

Step-by-Step Solution

Answer

The parametric equations are \\(x(t) = 1 + t, y(t) = -1 - 2t, z(t) = 2 + t\\).

Step 1: Understand the Problem

A parametric equation of a line in 3D space can be expressed using a point on the line and a direction vector. The point given is \(1, -1, 2\) and the direction vector is \[\begin{bmatrix} 1 \ -2 \ 1 \end{bmatrix}\]. We need to construct the parametric equation using these.

Step 2: Write the General Form of the Parametric Equation

A line in parametric form is given by \[ \begin{align*} x(t) &= x_0 + at \y(t) &= y_0 + bt \z(t) &= z_0 + ct \end{align*} \]Where \(x_0, y_0, z_0\) is a point on the line and \(a, b, c\) are the components of the direction vector.

Step 3: Substitute the Given Values

Substitute the point \(1, -1, 2\) for \(x_0, y_0, z_0\) and the vector \[1, -2, 1\] for \(a, b, c\) into the general form:\[ \begin{align*} x(t) &= 1 + 1t = 1 + t \y(t) &= -1 - 2t \z(t) &= 2 + t \end{align*} \]

Step 4: Write the Parametric Equations

The parametric equations of the line are:\[ \begin{align*} x(t) &= 1 + t \y(t) &= -1 - 2t \z(t) &= 2 + t \end{align*} \]These equations fully describe the line going through the point \(1, -1, 2\) in the given direction.

Key Concepts

3D Space

Imagine the world around you in three dimensions: length, width, and height. This is 3D space, which you encounter every day. In mathematics, we represent this space with three axes - typically labeled as the x-axis, y-axis, and z-axis. These axes intersect at a point called the origin, and extend infinitely in positive and negative directions. Any point in this 3D space can be described using coordinates \(x, y, z\).
For example, think of how a room is organized – it has length, width, and height. In math terms, if you wanted to describe a specific point in the room, you would say how far it is from the corner along each of these dimensions. These measurements correspond to the x, y, and z coordinates in 3D mathematical space.
When working with parametric equations, understanding 3D space is crucial because you'll often need to navigate through it to describe lines, shapes, or movements smoothly.

Direction Vector

A direction vector gives us an idea about where and how a line stretches or where it points in space. It acts like a compass showing the direction a line will follow.
In our example, the direction vector is \(\begin{bmatrix} 1 \ -2 \ 1 \end{bmatrix}\). This means:

• For every step in the positive x-direction, take one step.
• For every step in the negative y-direction, take two steps.
• For every step in the positive z-direction, take one step.

The numbers themselves act as multipliers for each axis, showing not just the direction, but also how slanted or steep the direction is. Think of it like arrows that dictate the trajectory from a start point. The magnitude and angle of this vector will influence where the line goes, whether it will rise, fall, or stay in place.

Point-Line Equation

To define a line uniquely in 3D space, you need a specific starting point and a direction vector. The point-line equation is a tool that combines these to give a clear road map of where the line exists in space.
Using the example, the specific point given is \(1, -1, 2\), and the direction vector is \(\begin{bmatrix} 1 \ -2 \ 1 \end{bmatrix}\). The general form of the parametric equation used to find the line is:
\(x(t) = x_0 + at, \ y(t) = y_0 + bt, \ z(t) = z_0 + ct\)
Here, \(x_0, y_0, z_0\) represents the starting point, and \(a, b, c\) represent the components of the direction vector. When you substitute the given values into these equations, you create a set of parametric equations that describe every point along the line:

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

These equations allow you to track or plot any desired point on the line as \(t\) varies. This is essential in computer graphics, animations, and physics simulations, where knowing the position of objects along a path in 3D space is necessary.