91Ó°ÊÓ

The derivative of a vector function involves differentiating each component of the vector separately. In the context of the exercise, we are given the vector function \( \mathbf{r}(t) = 2 \sin t \mathbf{i} + 3 \cos t \mathbf{j} \). To find the derivative \( \mathbf{r}^{\prime}(t) \), you must differentiate the \( \sin t \) and \( \cos t \) parts independently.

The differentiation process follows these steps:

• The derivative of \( 2 \sin t \) with respect to \( t \) is \( 2 \cos t \), as the derivative of \( \sin t \) is \( \cos t \).
• The derivative of \( 3 \cos t \) is \(-3 \sin t \), as the derivative of \( \cos t \) is \(-\sin t \).

Thus, \( \mathbf{r}^{\prime}(t) \) becomes \( 2 \cos t \mathbf{i} - 3 \sin t \mathbf{j} \). This derivative represents the rate of change of the vector function with respect to \( t \), showing how the position vector's direction and length are changing over time.

Parametric equations describe a set of related quantities as dependent on a parameter, in this case, \( t \). For the vector function \( \mathbf{r}(t) = 2 \sin t \mathbf{i} + 3 \cos t \mathbf{j} \), the parametric equations can be expressed as:

• \( x = 2 \sin t \)
• \( y = 3 \cos t \)

These equations define the coordinates \( (x, y) \) of a point on a curve as functions of \( t \).

As you evaluate these functions for different values of \( t \), you're essentially tracing the path of the curve, which is the main advantage of using parametric equations. They allow us to easily describe complex curves like ellipses or circles that would otherwise be more challenging to define with standard equations. By plugging in different \( t \) values, we can determine the specific (x, y) positions along the curve.

When sketching an ellipse defined by parametric equations, it's essential to understand the geometry described. With \( x = 2 \sin t \) and \( y = 3 \cos t \), these equations describe an ellipse centered at the origin. Here, the ellipse's semi-major and semi-minor axes are determined by the coefficients of the trigonometric functions.

For this particular ellipse:

• The semi-major axis is 3 (in the y-direction), due to the coefficient of \( \cos t \).
• The semi-minor axis is 2 (in the x-direction), due to the coefficient of \( \sin t \).

To sketch this curve, you can plot significant points, such as:

• \( t = 0 \), giving the point \( (0, 3) \)
• \( t = \frac{\pi}{2} \), giving the point \( (2, 0) \)
• \( t = \pi \), giving the point \( (0, -3) \)
• \( t = \frac{3\pi}{2} \), giving the point \( (-2, 0) \)

Once these points are plotted, you'll have a clear view of the elliptical path. Additionally, by finding the tangent vector at a specific point, you can better understand the curve's orientation and direction at any given instance. The tangent vector is critical for analyzing the direction and slope at a certain point along the ellipse.