91Ó°ÊÓ

In three-dimensional space, Cartesian coordinates are used to denote the position of a point using three values. These values represent distances from a set of perpendicular lines that intersect at an origin. In our case, the lines correspond to the x, y, and z axes.
To transform a parametric form like \( \mathbf{c}(t) = (2t - 1, t + 2, t) \) into Cartesian coordinates, we substitute the parameter \( t \) with one of the coordinates—in this instance, \( z = t \).
By substituting \( t = z \) in the other parametric equations, we determine:

• \( x = 2z - 1 \)
• \( y = z + 2 \)

These equations are now fully in Cartesian form, describing the position of a point on the curve in terms of \( x \), \( y \), and \( z \). This conversion is particularly useful for visualizing and sketching paths since Cartesian coordinates relate directly to the axes in three-dimensional graphs.

Vector functions are powerful tools in describing paths in 3D space. They capture the position of a point as a function of a single variable, typically \( t \), by using vector notation. For example, the vector function \( \mathbf{c}(t) = (2t - 1, t + 2, t) \) means that every value of \( t \) corresponds to a specific point \( (x, y, z) \) in space.
Each component of the vector function describes a relationship of one of the coordinates to the parameter \( t \). This helps in understanding how the point moves as \( t \) changes:

• \( x(t) = 2t - 1 \): Describes the path along the x-axis.
• \( y(t) = t + 2 \): Describes the path along the y-axis.
• \( z(t) = t \): Describes the path along the z-axis.

These components indicate that with each change in \( t \), the point traces out a smooth path in the 3D space. Understanding this allows us to convert between vector and Cartesian forms, making it easier to analyze and visualize 3D curves.

Sketching 3D curves involves translating mathematical equations into visual representations on a three-dimensional graph. Once you have the Cartesian equations, drawing these is straightforward with some understanding of geometry.
The path described by \( x = 2z - 1 \) and \( y = z + 2 \) forms a straight line, as each equation is linear. The simplicity of these equations means that as \( z \) increases or decreases, the path direction is controlled linearly by \( x \) and \( y \).
To sketch, evaluate the equations at specific values of \( z \), or equivalently \( t \). For example:

• When \( t = 0 \), \( (x, y, z) = (-1, 2, 0) \)
• When \( t = 1 \), \( (x, y, z) = (1, 3, 1) \)
• When \( t = -1 \), \( (x, y, z) = (-3, 1, -1) \)

Plot these points in a 3D space and connect them. The points will align, indicating a straight line. The ability to sketch 3D curves offers a tangible way to perceive the structure and progression of a vector function in three dimensions.