91Ó°ÊÓ

In vector calculus, understanding the concept of a position vector is crucial. A position vector is a vector that extends from the origin of a coordinate system to a point in space. For example, in our exercise, the position vector is given by \( \mathbf{r}(t) = \langle\sin t, \cos t, \cos t\rangle \). This represents the location in 3D space for each value of \( t \) within the interval from 0 to \( 2\pi \).

• It defines the path of a moving point (the trajectory).
• The coordinates within the vector describe its position in the space at any time \( t \).

The position vector is foundational for analyzing trajectories because it provides the information necessary to derive further concepts like velocity and acceleration.
In this case, despite the 3D nature of the position vector, the trajectory itself lies on a circle in \( \mathbb{R}^2 \) when considering the first two components. The identity \( \sin^2 t + \cos^2 t = 1 \) confirms that this path indeed forms a circle, illustrating the utility of the position vector in visualizing and understanding motion in space.

The velocity vector is another fundamental concept in vector calculus, derived from the position vector. It represents the rate of change of the position vector with respect to time, effectively describing the speed and direction of movement at any given point along the path.
For this particular problem, the velocity vector is obtained by differentiating the position vector \( \mathbf{r}(t) = \langle\sin t, \cos t, \cos t\rangle \) with respect to \( t \), resulting in \( \mathbf{v}(t) = \langle\cos t, -\sin t, -\sin t\rangle \).

• The velocity vector points tangentially to the trajectory at any point.
• It provides critical insight into how the position changes over time, showing both the magnitudes of speed and changes in direction.

Understanding how the velocity vector behaves allows us to explore concepts like acceleration and orthogonality, as seen when confirming orthogonal relationships between position and velocity vectors at all points along their path.

Orthogonality is a key concept in vector calculus that describes the perpendicularity of vectors. Two vectors are orthogonal if their dot product is zero.
In our exercise, we verify the orthogonality between the position vector \( \mathbf{r}(t) \) and the velocity vector \( \mathbf{v}(t) \) by computing their dot product: \[ \mathbf{r}(t) \cdot \mathbf{v}(t) = \langle\sin t, \cos t, \cos t\rangle \cdot \langle\cos t, -\sin t, -\sin t\rangle = 0 \]

• This zero result indicates that the position and velocity vectors are indeed orthogonal at every point \( t \).
• Orthogonality implies that the velocity is tangential and therefore indicative of circular motion when related to the position vector.

This property simplifies understanding motion along a path, as perpendicular vectors imply no component of one vector affects the magnitude of the other in the direction of the path.

Trigonometric identities are mathematical equations involving trigonometric functions that hold true for all values of the variables involved.
In this context, the trigonometric identity \( \sin^2 t + \cos^2 t = 1 \) plays a critical role. It helps validate that the trajectory defined by the position vector in the exercise lies on a circle:

• This identity ensures the sum of the squared sine and cosine values remains constant, indicating a continuous circular path with a fixed radius.
• Understanding these identities allows us to simplify the problem, quickly verifying the nature of trajectories in relation to circles and spheres.

Furthermore, applying such identities prepares students for more complex engagements with vector calculus. Mastery of these foundational identities supports solving a wide range of problems in mathematics and physics.