91Ó°ÊÓ

The concept of a position vector is crucial in understanding motion within a coordinate system. In three-dimensional space, a position vector identifies a point or the end of a path made by a moving object relative to the origin of the space. For the journey described in this exercise, the position vector is given by \( \mathbf{r}(t) = (3 \cos t) \mathbf{i} + (3 \sin t) \mathbf{j} + t^{2} \mathbf{k} \). Here, the components of the vector \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) represent the unit directions corresponding to the x, y, and z axes respectively.
The essence of this vector is that it plays a pivotal role in pinpointing the hang glider's exact location as time progresses. Each of its components - \( 3 \cos t \), \( 3 \sin t \), and \( t^2 \) - specifies how the x, y, and z coordinates change over time. This paints a complete picture of the path in three dimensions.

When examining the path of the hang glider in the xy-plane, the components \( x(t) = 3 \cos t \) and \( y(t) = 3 \sin t \) illustrate circular motion. This type of motion occurs when an object moves along a path in the shape of a circle, consistently maintaining a specific radius from a central point.
In our scenario:

• The center of the circle is at the origin (0, 0),
• The radius of the circle is 3, as both the \( x \) and \( y \) expressions are multiplied by 3,
• The path traced creates a consistent loop as \( t \) changes, maintaining a rotational motion.

While this explains motion only on the xy-plane, it demonstrates how the hang glider continuously follows a looping path within that plane.

The upward motion of the hang glider isn't just straightforward; it involves a precise quadratic nature as time \( t \) progresses. The z-component, represented by \( z(t) = t^2 \), introduces quadratic upward motion, which essentially suggests a type of acceleration.

This concept can be further understood by analyzing why this motion is called 'quadratic':

• As time increases, the square of \( t \) grows larger at a faster rate,
• This results in a rise in height that accelerates over time, not moving uniformly as it would with a linear function,
• Thus, the hang glider ascends ever faster in the z-direction, differentiating the path from a standard spiral where z increases at a fixed rate.

This non-linear climb distinguishes the path from a helix, making the motion unique and dynamic as the hang glider continually gains altitude.