Problem 58 - 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. $$ (2,1,-3),\left[\begin{array}{r} 3 \\ -1 \\ 2 \end{array}\right] $$

Step-by-Step Solution

Answer

The parametric equations are \( x(t) = 2 + 3t \), \( y(t) = 1 - t \), and \( z(t) = -3 + 2t \).

Step 1: Understand the Line Equation

A line in space can be described using a point and a direction vector. The general parametric form of a line in 3D space is given by the equation: \( \mathbf{r}(t) = \mathbf{a} + t \mathbf{b} \), where \( \mathbf{a} \) is a point on the line, \( \mathbf{b} \) is a direction vector, and \( t \) is a parameter.

Step 2: Identify the Point and Direction Vector

The given point through which the line passes is \( (2, 1, -3) \), and the direction vector is \( \begin{bmatrix} 3 \ -1 \ 2 \end{bmatrix} \). This means the components \( x, y, z \) of the line's equations can be individually expressed.

Step 3: Write the Parametric Equations

Using the point \( (2, 1, -3) \) and direction vector \( \begin{bmatrix} 3 \ -1 \ 2 \end{bmatrix} \), write the parametric equations for each coordinate:1. \( x(t) = 2 + 3t \)2. \( y(t) = 1 - t \)3. \( z(t) = -3 + 2t \).

Key Concepts

Exploring 3D Space

Welcome to the fascinating world of three-dimensional (3D) space! In 3D space, we visualize objects with depth, providing a richer insight into their placement. Every point can be described by three coordinates, \(x, y, z\), representing the location on each axis.
3D space is crucial in areas like computer graphics, physics, and engineering. Here, each point in space is unique and determined by going along three perpendicular lines - the x, y, and z axes.

• x-axis: Represents horizontal movement
• y-axis: Represents vertical movement
• z-axis: Represents depth

Understanding these axes helps in visualizing lines and shapes in the physical world. They are used to calculate distances, angles, and much more, allowing precise modeling of real-world scenarios.

Direction Vector

Direction vectors play a key role when defining lines and transformations. A direction vector points from one location to another and indicates the direction and magnitude of movement.

In the context of a line, the direction vector shows the line's orientation in space. For instance, the vector \([3, -1, 2]\) suggests the line moves; in positive x, negative y, and positive z directions. Observing the signs and values helps identify how steeply each line component changes.

• Magnitude: Length of the vector, calculated using the Pythagorean theorem in 3D: \(|\mathbf{b}| = \sqrt{3^2 + (-1)^2 + 2^2} = \sqrt{14}\)
• Components: The increments along each axis: 3 (x), -1 (y), and 2 (z)

Direction vectors simplify modeling paths in engineering and animations. Remember, the line's velocity is not altered by scaling the direction vector.

Line Equation

A line equation in 3D space describes a path using a specific starting point and a direction vector. Parametric equations powerfully define these lines, allowing calculations of any point on the line given a parameter.
The parametric form of a line equation is expressed as: \( \mathbf{r}(t) = \mathbf{a} + t \mathbf{b} \), where:

• \(\mathbf{a}\): A fixed point, e.g., \( (2, 1, -3) \)
• \(\mathbf{b}\): Direction vector, e.g., \( [3, -1, 2] \)
• \(t\): A scalar parameter that varies along the line

The parametric equations for each coordinate are:
\[x(t) = 2 + 3t, \ y(t) = 1 - t, \ z(t) = -3 + 2t\]
These separate equations imply that tweaking \(t\) traces every possible point along the line. Parametric line equations are instrumental in trajectory planning, navigation systems, and graphical applications.