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 equation is \((x, y, z) = (2 + 3t, 1 - t, -3 + 2t)\).

Step 1: Understanding the Problem

We need to find a parametric equation of a line that passes through a given point and is directed along a given vector. The point provided is \((2, 1, -3)\) and the direction vector given is \(\begin{bmatrix} 3 \ -1 \ 2 \end{bmatrix}\).

Step 2: Formulating the Parametric Equation

The parametric equation of a line in 3D space can be written in the form \((x, y, z) = (x_0, y_0, z_0) + t(a, b, c)\), where \((x_0, y_0, z_0)\) is the given point, \((a, b, c)\) is the direction vector, and \(t\) is a parameter. Here, \((x_0, y_0, z_0) = (2, 1, -3)\) and \(\begin{bmatrix} a \ b \ c \end{bmatrix} = \begin{bmatrix} 3 \ -1 \ 2 \end{bmatrix}\).

Step 3: Writing the Parametric Equations

Substitute the point and direction vector into the parametric equation formula: \((x, y, z) = (2, 1, -3) + t(3, -1, 2)\). This results in the parametric equations:\[x = 2 + 3t\]\[y = 1 - t\]\[z = -3 + 2t\]

Step 4: Final Parametric Equation

The parametric equation of the line that passes through the point \((2, 1, -3)\) and follows the direction vector \(\begin{bmatrix} 3 \ -1 \ 2 \end{bmatrix}\) is:\[ (x, y, z) = (2 + 3t, 1 - t, -3 + 2t) \]

Key Concepts

3D Coordinate Geometry

In the realm of 3D coordinate geometry, we delve into the study of spatial relationships and figures in three-dimensional space. Unlike two-dimensional geometry, which deals with flat surfaces, 3D coordinate geometry adds another dimension to include depth. This makes it not only fascinating but also essential for solving complex real-world problems.

In 3D coordinate geometry:

• Points are represented as ordered triples, such as \( (x, y, z) \), where each coordinate corresponds to a position along the x-, y-, and z-axes.
• Lines in 3D space can be described using vector equations or parametric equations.
• Planes are defined usually through an equation involving x, y, and z, generally leading to solutions involving all three coordinates simultaneously.

Understanding 3D geometry is crucial because it reflects the physical realm in which we live and enables applications ranging from computer graphics to physics and engineering.

Vector Direction

A vector in space is a quantity that has both magnitude and direction. It is represented in the form \( \begin{bmatrix} a \ b \ c \end{bmatrix} \), where \(a, b, \) and \(c\) denote the vector's components along the x-, y-, and z-axes, respectively.

When it comes to understanding the direction of a vector:

• The direction is essentially the orientation of how the vector points in space.
• The components \( a, b, \) and \( c \) help determine this direction relative to each axis.
• Direction vectors are crucial for describing lines and motions in space, providing a roadmap for where and how something moves or aligns.

Using the given vector \( \begin{bmatrix} 3 \ -1 \ 2 \end{bmatrix} \), for instance, it tells us which way to head from the initial point. It says to move three units along the x-axis, one unit backwards along the y-axis, and two units forward along the z-axis. This sequentially gives us a vivid picture of the line's path in 3D space.

Parametric Equations in Space

Parametric equations in space provide a way to describe curves and lines by expressing their coordinates as functions of one or more parameters. This is particularly useful in 3D geometry for articulating lines or paths in space that aren't easily described using simple equations.

With parametric equations:

• We often employ a parameter, like \( t \), which varies over the range of the equation to trace out a line or a curve.
• A standard equation of a point on a line is given by \( (x, y, z) = (x_0, y_0, z_0) + t(a, b, c) \), where \( t \) is the parameter, \( (x_0, y_0, z_0) \) is a point on the line, and \( (a, b, c) \) is the direction vector.
• Adjusting the parameter \( t \) effectively moves the point along the line, starting at \( (x_0, y_0, z_0) \) and following the vector's direction.

For example, the equation derived earlier \[ (x, y, z) = (2 + 3t, 1 - t, -3 + 2t) \] describes a line in 3D space where each value of \( t\) results in new coordinates along the described line, allowing us to visualize and understand the spatial path comprehensively.