91Ó°ÊÓ

When we talk about a vector function, envision it as a machine which takes numbers as inputs and puts out vectors as outputs. In more formal language, it's a function that assigns a vector to each input value from its domain, most commonly a subset of real numbers. For example, in our exercise, the vector function is defined as \( \mathbf{r}(t)=e^{16 t} \cos t \mathbf{i}+e^{16 t} \sin t \mathbf{j}+e^{16 t} \mathbf{k} \).

Here, each component of the vector—associated with \( \mathbf{i} \) (representing the x-direction), \( \mathbf{j} \) (the y-direction), and \( \mathbf{k} \) (the z-direction)—is a function of the variable \( t \). As \( t \) changes, each component of the vector \( \mathbf{r}(t) \) also changes, tracing out a curve in three-dimensional space. This kind of function is incredibly useful in physics and engineering to describe paths or trajectories of particles and objects through space over time.

The concept of derivative of a vector function extends the idea of derivatives from calculus to functions that output vectors instead of just numbers. If our vector function represents the position of a particle over time, the derivative of this function with respect to time \( t \) would give us the velocity of the particle at any given moment.

For the derivative of a vector function, you differentiate its components individually with respect to the variable. In the exercise, we calculate \( \mathbf{r'}(t) \) by taking the derivatives of the function's \( \mathbf{i} \) , \( \mathbf{j} \) , and \( \mathbf{k} \) components separately, applying the chain rule and the product rule where necessary. The derivative \( \mathbf{r'}(t) \) provides the instantaneous rate of change of the vector function, which can be related to many physical aspects such as velocity, if the original function \( \mathbf{r}(t) \) represents position.

To normalize a vector means to scale the vector so that it has a length (also referred to as magnitude) of 1 without changing its direction. This process involves dividing the vector by its own magnitude. Normalized vectors are often called unit vectors and are especially useful because they maintain directionality while simplifying calculations.

As we saw in the exercise, normalizing the derivative \( \mathbf{r'}(t) \) at the given point \( t = \frac{\pi}{2} \) provides us with the unit tangent vector \( \mathbf{T}(\frac{\pi}{2}) \) at that point on the curve. The tangent vector gives the direction of the curve at a certain point and, when normalized, is crucial in computations involving curvature, arc length, and other vector functions in space.