91Ó°ÊÓ

Parametric equations are a way to describe a curve by expressing the coordinates that define the curve as functions of a single variable, usually denoted as \( t \).
In the context of the exercise, the vector-valued function \( \mathbf{F}(t) = \cos t \mathbf{i} + t \mathbf{j} - \sin t \mathbf{k} \) consists of three parametric equations:

• \( x(t) = \cos t \)
• \( y(t) = t \)
• \( z(t) = -\sin t \)

These equations separately describe how each coordinate (\( x \), \( y \), and \( z \)) changes as \( t \) varies from 0 to \( 2\pi \).
This allows us to understand the movement of a point along the curve by observing how it progresses in each dimension independently.

In 3-dimensional space, we work with three axes: the x-axis, y-axis, and z-axis.
Each axis represents a different direction in space, and together they form a coordinate system that can describe any point within that space.
A vector-valued function, like the one given in the exercise, represents a curve by associating each point on the curve with a particular value of \( t \).
In the exercise \( \mathbf{F}(t) = \cos t \mathbf{i} + t \mathbf{j} - \sin t \mathbf{k} \), we see:

• \( \cos t \) indicating movement along the x-axis
• \( t \) guiding motion along the y-axis
• \(-\sin t \) driving changes along the z-axis

By plotting these changes, we create a visual representation of the curve in 3-dimensional space, offering insights into how it twists and turns as each coordinate value is affected by the changing parameter \( t \).

Curve sketching using parametric equations involves plotting points at various values of \( t \) and connecting them smoothly.
For the given function, we seek key points that highlight the path of the curve.
For example, by choosing \( t = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, \text{and} 2\pi \), we determine specific values for each coordinate:

• At \( t = 0 \), we have: \( (x, y, z) = (1, 0, 0) \)
• \( t = \frac{\pi}{2} \): \( (x, y, z) = (0, \frac{\pi}{2}, -1) \)
• \( t = \pi \): \( (x, y, z) = (-1, \pi, 0) \)
• \( t = \frac{3\pi}{2} \): \( (x, y, z) = (0, \frac{3\pi}{2}, 1) \)
• \( t = 2\pi \): \( (x, y, z) = (1, 2\pi, 0) \)

Plotting these points in sequence and joining them provides a rough sketch of the curve, demonstrating the cyclic nature due to \( \cos t \) and \( -\sin t \), combined with a steady increase along the y-axis.

Analyzing the behavior of a vector-valued function helps us understand the characteristics and movement pattern of the curve.
In the vector function \( \mathbf{F}(t) = \cos t \mathbf{i} + t \mathbf{j} - \sin t \mathbf{k} \):

• \( x(t) = \cos t \) oscillates between -1 and 1 as \( t \) progresses, creating a wave-like motion along the x-axis.
• \( y(t) = t \) increases linearly, suggesting a consistent upward or forward direction along the y-axis. This linearity implies the curve never turns back in the y-direction.
• \( z(t) = -\sin t \) moves from 0 to -1 and back to 0 within each cycle of \( \pi \), adding vertical oscillation along the z-axis.

These components combined produce a helical path, rising along the y-axis while oscillating in the x and z dimensions.
Understanding this behavior allows for better prediction of the curve's path and its directional qualities.