Problem 24 - Advanced Engineering Mathematics

Question

Graph the curve \(C\) that is described by \(\mathbf{r}\) and graph \(\mathbf{r}^{\prime}\) at the indicated value of \(\boldsymbol{t}\). $$ \mathbf{r}(t)=3 \cos t \mathbf{i}+3 \sin t \mathbf{j}+2 t \mathbf{k} ; t=\pi / 4 $$

Step-by-Step Solution

Answer

Graph the curve of the helix and the tangent vector at \(t = \pi/4\) with computed points and vectors.

Step 1: Understand the Vector Function

We are given the vector-valued function \(\mathbf{r}(t) = 3 \cos t \mathbf{i} + 3 \sin t \mathbf{j} + 2t \mathbf{k}\). This function describes a curve in 3D space, with the parameters representing movements along the x, y, and z axes. The terms \(3 \cos t\) and \(3 \sin t\) represent a circle in the xy-plane, while \(2t\) is a linear function affecting the z-axis, which means the path is a helix.

Step 2: Find the Derivative of the Vector Function

To graph the tangent vector \(\mathbf{r}^{\prime}(t)\), we first need to differentiate \(\mathbf{r}(t)\) with respect to \(t\).- The derivative of \(3 \cos t\) is \(-3 \sin t\).- The derivative of \(3 \sin t\) is \(3 \cos t\).- The derivative of \(2t\) is \(2\).As such, \(\mathbf{r}^{\prime}(t) = -3 \sin t \mathbf{i} + 3 \cos t \mathbf{j} + 2 \mathbf{k}\).

Step 3: Compute \(\mathbf{r}(\pi / 4)\) and \(\mathbf{r}^{\prime}(\pi / 4)\)

Now, substitute \(t = \pi / 4\) into both \(\mathbf{r}(t)\) and \(\mathbf{r}^{\prime}(t)\).- For \(\mathbf{r}(\pi / 4)\): \(x = 3 \cos (\pi / 4) = 3 \times \frac{\sqrt{2}}{2}\) \(y = 3 \sin (\pi / 4) = 3 \times \frac{\sqrt{2}}{2}\) \(z = 2 \times \pi / 4 = \frac{\pi}{2}\)- For \(\mathbf{r}^{\prime}(\pi / 4)\): \(x' = -3 \sin (\pi / 4) = -3 \times \frac{\sqrt{2}}{2}\) \(y' = 3 \cos (\pi / 4) = 3 \times \frac{\sqrt{2}}{2}\) \(z' = 2\)Thus, \(\mathbf{r}(\pi / 4) = \left(\frac{3\sqrt{2}}{2}, \frac{3\sqrt{2}}{2}, \frac{\pi}{2}\right)\) and \(\mathbf{r}^{\prime}(\pi / 4) = \left(-\frac{3\sqrt{2}}{2}, \frac{3\sqrt{2}}{2}, 2\right)\).

Step 4: Graph the Curve and Tangent Vector

The graph of the curve \(\mathbf{r}(t)\) can be visualized as a helix starting from the xy-plane and spiraling upward as \(t\) increases. At the point \(\mathbf{r}(\pi / 4)\), draw the tangent vector \(\mathbf{r}^{\prime}(\pi / 4)\), starting at this point. The tangent vector will indicate the direction in which the curve is moving at \(t = \pi / 4\). This vector should be drawn proportionately at the point \(\left(\frac{3\sqrt{2}}{2}, \frac{3\sqrt{2}}{2}, \frac{\pi}{2}\right)\).

Key Concepts

Vector Functions

Vector functions are a powerful tool in mathematics used to describe paths or curves in 2D and 3D space. A vector function assigns a vector to each point of a domain, typically using a parameter (commonly denoted by \(t\)) to trace out the curve or path as it changes over time.
For example, in the vector function \(\mathbf{r}(t) = 3 \cos t \mathbf{i} + 3 \sin t \mathbf{j} + 2t \mathbf{k}\), each value of \(t\) depicts a point in three-dimensional space, giving us a full curve as \(t\) ranges over its domain.
- The component \(3 \cos t \mathbf{i}\) determines the x-coordinate.
- The component \(3 \sin t \mathbf{j}\) determines the y-coordinate.
- The component \(2t \mathbf{k}\) determines the z-coordinate.
By combining these components, vector functions allow us to visualize intricate 3D figures, providing a detailed path for objects or phenomena in space.

Helix

A helix is a fascinating geometrical shape, often described as a "corkscrew" or spiral. It is defined by having a constant radius and a constant gap between the turns in the z-direction. Helices appear in various fields, such as biology (DNA strands) and engineering (springs and screws).
In mathematical terms, a helix is often expressed through a vector function where the x and y components form a circle, while the z component increases linearly with the parameter \(t\).
- Example: The vector function \(\mathbf{r}(t) = 3 \cos t \mathbf{i} + 3 \sin t \mathbf{j} + 2t \mathbf{k}\) describes a helix.
- The terms \(3 \cos t\) and \(3 \sin t\) create a circular motion in the xy-plane.
- The term \(2t\) ensures a steady rise along the z-axis, creating the spiral shape in 3D.
Recognizing a curve as a helix helps in visualizing its 3D structure, enhancing our understanding of motion and design in space.

Tangent Vectors

Tangent vectors provide insight into the direction and rate of change of a vector function at any given point along its path. They are akin to the concept of derivatives in calculus, but applied to vector functions in space.
A tangent vector can be found by differentiating the vector function with respect to its parameter \(t\). For our function \(\mathbf{r}(t) = 3 \cos t \mathbf{i} + 3 \sin t \mathbf{j} + 2t \mathbf{k}\), the derivative \(\mathbf{r}^{\prime}(t) = -3 \sin t \mathbf{i} + 3 \cos t \mathbf{j} + 2 \mathbf{k}\) represents the tangent vector.

• The x-component \(-3 \sin t\) indicates how the position along the x-axis changes with \(t\).
• The y-component \(3 \cos t\) shows the change along the y-axis.
• The z-component \(2\) shows a steady change in the z-direction.

At any point \(t\), the tangent vector helps determine the instantaneous direction in which the curve continues, proving crucial in understanding motion and dynamics.

3D Graphing

3D graphing is the process of plotting points, lines, and curves in three-dimensional space. This method reveals the full spatial structure of paths and shapes that vector functions describe.
To graph a curve like \(\mathbf{r}(t) = 3 \cos t \mathbf{i} + 3 \sin t \mathbf{j} + 2t \mathbf{k}\), you follow these simple steps:

• Map out individual points for discrete values of \(t\).
• Connect these points smoothly to visualize the continuous path.
• Use tools, like graphing calculators or software, to handle complex shapes.

Additionally, indicating tangent vectors at key points allows you to convey the direction of the path vividly. Graphing such 3D structures assists in visualizing concepts in physics, engineering, and beyond, enhancing comprehension and communication of spatial data.