91Ó°ÊÓ

In vector calculus, understanding the angle between vectors is crucial as it can indicate how these vectors relate to one another in space. The angle, denoted as \( \theta(t) \), between two vectors \( \mathbf{r}(t) \) and \( \mathbf{r}'(t) \) can be determined using the dot product. This fundamental operation involves multiplying corresponding components of the vectors and summing up these products. Mathematically, it is given by:

• \( \mathbf{r}(t) \cdot \mathbf{r}'(t) = (4\cos t)(-4\sin t) + (3\sin t)(3\cos t) = -7 \cos t \sin t. \)

To find the angle, we use:

• \( \cos \theta(t) = \frac{\mathbf{r}(t) \cdot \mathbf{r}'(t)}{|\mathbf{r}(t)| |\mathbf{r}'(t)|}. \)

By calculating \( \theta(t) \) over a set interval, we can visualize how the directional relationship between \( \mathbf{r}(t) \) and its derivative changes with respect to \( t \). This is essential in fields like physics and engineering, where understanding vector interactions is key to analyzing motion and forces.

Vector functions are a way to describe spatial relationships or paths of motion, using each component of the function to define a position over time or another variable. For instance, a vector function \( \mathbf{r}(t) = 4\cos t \mathbf{i} + 3\sin t \mathbf{j} \) traces a path in two-dimensional space. Here, \( \mathbf{i} \) and \( \mathbf{j} \) are the unit vectors in the direction of the x-axis and y-axis, respectively.

• The coefficient expressions \( 4\cos t \) and \( 3\sin t \) determine the x and y coordinates of the vector at any given time \( t \).
• Such functions are useful in modeling real-life scenarios where an object moves along a trajectory, influenced by varying forces or speeds.

By analyzing these functions, one can extract important information about motion, such as velocity and acceleration, by computing their derivatives.

Orthogonality is a concept that describes when vectors are perpendicular to each other. In terms of the vectors \( \mathbf{r}(t) \) and \( \mathbf{r}'(t) \), they are orthogonal if their dot product equals zero:

• \( \mathbf{r}(t) \cdot \mathbf{r}'(t) = 0. \)

In the context of this problem, finding \( t \) such that \( \theta(t) = 0 \) indicates orthogonality. This suggests that at those points, the original vector is moving in a direction perpendicular to its instantaneous rate of change. This property is often leveraged in computer graphics and physics simulations to manage orientation changes and to calculate forces in a balanced system.

The derivative of a vector function defines how the vector itself changes along its path. For an original vector function \( \mathbf{r}(t) = 4\cos t \mathbf{i} + 3\sin t \mathbf{j} \), its derivative \( \mathbf{r}'(t) \) provides the instantaneous rate of change, indicating the direction and velocity at any time \( t \).

• Calculate each component separately: \( \frac{d}{dt}(4\cos t) = -4\sin t \) and \( \frac{d}{dt}(3\sin t) = 3\cos t. \)
• The derivative \( \mathbf{r}'(t) = -4\sin t\mathbf{i} + 3\cos t\mathbf{j} \) gives the tangent's direction to the curve described by \( \mathbf{r}(t) \).

Evaluating the derivative enables critical insights into not only the path of a vector function but also the behavior of systems where speed, stability, or oscillatory patterns are of interest. In practical applications, this can translate to understanding velocity in the physical world or interpolating data in computations.