91Ó°ÊÓ

Parametric equations are a powerful mathematical tool to describe paths and curves in space. Unlike regular equations that relate two variables directly, parametric equations use one or more parameters, typically denoted as \( t \), to define the coordinates of a point in terms of functions of \( t \).

In the context of the given exercise, the vector-valued function \( \mathbf{r}(t) = \left(50 e^{-t} \cos t\right) \mathbf{i}+\left(50 e^{-t} \sin t\right) \mathbf{j}+\left(5-5 e^{-t}\right) \mathbf{k} \) is a set of parametric equations for a curve in three-dimensional space.

Here, the parameter \( t \) influences how the point on the curve shifts over time. This means as \( t \) changes, the individual components of the vector \( \mathbf{r}(t) \) trace different paths in the \( \mathbf{i}, \mathbf{j}, \) and \( \mathbf{k} \) directions, each calculated as a function of \( t \). Thus, parametric equations allow for a flexible representation of complex geometries and motions.

Vector calculus comes into play when dealing with vector-valued functions, like the one in our exercise.

These functions map a single parameter to vector quantities, ideal for describing physical magnitudes in fields like physics and engineering.

In the vector-valued function \( \mathbf{r}(t) = \left(50 e^{-t} \cos t\right) \mathbf{i}+\left(50 e^{-t} \sin t\right) \mathbf{j}+\left(5-5 e^{-t}\right) \mathbf{k} \), each component represents a direction in space, typically denoted with \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) units.

Using vector calculus, we can compute the derivative of \( \mathbf{r}(t) \) to find the tangent vector, allowing us to analyze the rate of change of the position along the curve.

This can provide insights into the properties of a curve or path, such as identifying points where it has a maximum or minimum turning point or accelerating.

The initial point of a path described by a vector-valued function is simply the position of the object or entity when the parameter \( t \) is equal to zero.

It's a significant value as it serves as a reference or starting point for understanding how a system evolves over time.

In the example \( \mathbf{r}(t) = \left(50 e^{-t} \cos t\right) \mathbf{i}+\left(50 e^{-t} \sin t\right) \mathbf{j}+\left(5-5 e^{-t}\right) \mathbf{k} \), finding \( \mathbf{r}(0) \) tells us where the point begins its journey.

By calculating each component with \( t \) set to zero, we determine that the initial point of this specific path is \( (50, 0, 0) \).

Knowing the initial point provides clarity, offering a clear marker from which subsequent positions can be traced or calculated as \( t \) varies.