Problem 18 - Calculus III

Question

Find the unit tangent vector for the following parameterized curves.\(\mathbf{r}(t)=t \mathbf{i}+3 t \mathbf{j}+t^{2} \mathbf{k}\)

Step-by-Step Solution

Answer

The unit tangent vector is \( \frac{\mathbf{i} + 3\mathbf{j} + 2t\mathbf{k}}{\sqrt{10 + 4t^2}} \).

Step 1: Find the Derivative of the Vector Function

The first step is to find the derivative of the vector function \( \mathbf{r}(t) = t \mathbf{i} + 3t \mathbf{j} + t^2 \mathbf{k} \). Calculate \( \frac{d\mathbf{r}}{dt} \). The derivative is \( \mathbf{r}'(t) = \frac{d}{dt}(t) \mathbf{i} + \frac{d}{dt}(3t) \mathbf{j} + \frac{d}{dt}(t^2) \mathbf{k} = \mathbf{i} + 3\mathbf{j} + 2t\mathbf{k} \).

Step 2: Calculate the Magnitude of the Derivative

Next, calculate the magnitude of \( \mathbf{r}'(t) = \mathbf{i} + 3\mathbf{j} + 2t\mathbf{k} \). The magnitude is \( \| \mathbf{r}'(t) \| = \sqrt{1^2 + 3^2 + (2t)^2} = \sqrt{1 + 9 + 4t^2} = \sqrt{10 + 4t^2} \).

Step 3: Divide the Derivative by its Magnitude

Finally, divide the derivative \( \mathbf{r}'(t) \) by its magnitude to find the unit tangent vector. The unit tangent vector is \( \mathbf{T}(t) = \frac{\mathbf{r}'(t)}{\| \mathbf{r}'(t) \|} = \frac{\mathbf{i} + 3\mathbf{j} + 2t\mathbf{k}}{\sqrt{10 + 4t^2}} \).

Key Concepts

Understanding Parameterized Curves

A parameterized curve is a type of mathematical representation that uses a parameter, typically denoted by \( t \), to describe a curve. In contrast to using just \( x \) and \( y \) coordinates, parameterized curves leverage a "time-like" parameter which can vary independent of the actual position along the curve. This provides a flexible way to describe complex shapes.

- **Vector Representation**: Curves are expressed using vector functions, such as \( \mathbf{r}(t) = t \mathbf{i} + 3t \mathbf{j} + t^2 \mathbf{k} \). Here, \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) are unit vectors in the direction of the axes.
- **Point Representation:** At each point \( t \), \( \mathbf{r}(t) \) gives the position of the point on the curve. As \( t \) changes, the point moves along the curve.

Parameterized curves are particularly useful when dealing with motion and dynamics, as they can easily express positions over time. They serve as the foundation for understanding vector calculus concepts, such as derivatives.

Exploring the Vector Function Derivative

The derivative of a vector function provides critical insights into how a curve changes. It shows the direction and rate of change at any given point on the parameterized curve.

- **Computing the Derivative**: For a vector function \( \mathbf{r}(t) = t \mathbf{i} + 3t \mathbf{j} + t^2 \mathbf{k} \), the derivative is calculated component-wise: \( \mathbf{r}'(t) = \mathbf{i} + 3\mathbf{j} + 2t\mathbf{k} \). This involves differentiating each component separately: \( \frac{d}{dt}(t) = 1 \), \( \frac{d}{dt}(3t) = 3 \), and \( \frac{d}{dt}(t^2) = 2t \).
- **Resulting Vector**: The resulting derivative vector still lives in the same vector space. It represents the velocity (tangent) vector at each point \( t \) on the curve.

Understanding vector derivatives not only helps in grasping how movement along a path occurs but also lays the groundwork for comprehending integral calculus within vector spaces.

Calculating the Magnitude of a Vector

The magnitude of a vector, sometimes referred to as its length or norm, conveys how "large" a vector is regardless of its direction. Calculating this is essential when you need to scale vectors for obtaining unit vectors.

- **Magnitude Formula**: Given a vector \( \mathbf{v} = a \mathbf{i} + b \mathbf{j} + c \mathbf{k} \), its magnitude is found using \( \| \mathbf{v} \| = \sqrt{a^2 + b^2 + c^2} \).
- **Example Calculation**: For \( \mathbf{r}'(t) = \mathbf{i} + 3\mathbf{j} + 2t\mathbf{k} \), the magnitude is calculated as \( \| \mathbf{r}'(t) \| = \sqrt{1 + 9 + (2t)^2} = \sqrt{10 + 4t^2} \).

Magnitudes are vital when normalizing vectors. Converting vectors to unit vectors, which have a magnitude of 1, is a crucial step in many physical applications, such as calculating the unit tangent vector, where you scale the vector derivative by its magnitude.