Question

Consider the curve. $$\mathbf{r}(t)=\langle a \cos t+b \sin t, c \cos t+d \sin t, e \cos t+f \sin t\rangle$$ where \(a, b, c, d, e,\) and \(f\) are real numbers. It can be shown that this curve lies in a plane. Graph the following curve and describe it. $$r(t)=(2 \cos t+2 \sin t) i+(-\cos t+2 \sin t) \mathbf{j}+(\cos t-2 \sin t) \mathbf{k}$$

Short answer

Question: Given the parametric curve $$\mathbf{r}(t) = (2\cos{t} + 2\sin{t}) \mathbf{i} + (-\cos{t} + 2\sin{t}) \mathbf{j} + (\cos{t} - 2\sin{t}) \mathbf{k},$$ describe the shape of the curve and its properties. Answer: The curve represents a non-uniform helix with a varying pitch and radius over the range of t values from 0 to \(2\pi\). The coefficients of the trigonometric functions dictate the planar representation and orientation changes of the helix in 3D space.

Step-by-step solution

Identify the vector components

We are provided with the given curve, \(\mathbf{r}(t)\), in parametric form: $$\mathbf{r}(t) = (2\cos{t} + 2\sin{t}) \mathbf{i} + (-\cos{t} + 2\sin{t}) \mathbf{j} + (\cos{t} - 2\sin{t}) \mathbf{k}.$$ We can identify the given vector components as: $$ a = 2, \quad b = 2,\\ c = -1, \quad d = 2,\\ e = 1, \quad f = -2. $$

Calculate the position of the curve for a range of t values

Now let's calculate the x, y, and z positions for a range of t values between \(0\) and \(2\pi\). We can do this by plugging in the values of t and calculating the x, y, and z coordinates using a set of t values interpolated in the range of \(0\) to \(2\pi\).

Graph the curve

Create a 3D plot of the curve using the x, y, and z coordinates calculated in step 2 for the different t values. As we plot these coordinates, we will get a visual shape of the curve in 3D space.

Describe the curve

The curve can be described as a helix with certain properties. As the coefficients of the trigonometric functions are different, the curve represents a helix that is non-uniform and has a varying pitch and radius over the range of t values. The position vectors a, b, c, d, e, and f dictate the planar representation and orientation changes of the helix in 3D space.

Key concepts

Vector Components

Understanding vector components is crucial when working with curves in three-dimensional space. A vector in 3D space is an arrow that has both a direction and a magnitude and is typically used to represent a point relative to an origin. It's composed of components along the x, y, and z axes, which are represented as \textbf{i}, \textbf{j}, and \textbf{k} respectively in the Cartesian coordinate system.

In the context of the exercise, the vector components are given by the parametric equations associated with each axis. For the curve defined by the vector function \(\textbf{r}(t)\), the components along the \textbf{i}, \textbf{j}, and \textbf{k} directions are functions of a trigonometric function of \(t\). Each component function pairs a cosine with a sine function, each multiplied by different constants that determine the direction and scaling of the vector in space. By identifying these components, we can analyze the curve's behavior in each dimension separately, which is an initial step to visualizing and understanding the shape and position of the curve in three-dimensional space.

3D Curve Graphing

Graphing a curve in three dimensions involves plotting points in space defined by parametric equations over a range of values. In the exercise, the curve is defined by a set of parametric equations, each representing a vector component as a function of \(t\). To create a 3D graph, you would typically calculate the x, y, and z coordinates for a series of values of \(t\) and then mark these points in a 3D coordinate system.

Software tools or graphing calculators are often used for this purpose since they can handle the calculations and the drawing scale effectively. As the values of \(t\) are varied from \(0\) to \(2\textbackslash{\backslash pi}\), the respective coordinates trace out the path of the curve in space. The resulting graph allows us to see the actual path that the curve takes, including its orientation and how it changes direction and curvature.

Trigonometric Functions

Trigonometric functions are fundamental in studying the periodic phenomena, such as cycles, waves, and circular motion, which naturally extend to the analysis of curves in calculus. Functions like sine and cosine describe relationships between the angles and lengths of triangles but they also model periodic motion.

In our parametric equations, the sine and cosine functions take the parameter \(t\), which could represent time or an angle, as their input. The resulting output of these functions modulates the curve, affecting its height and position as \(t\) varies. The presence of trigonometric functions indicates that the curve will have a periodic nature, potentially creating loops or waves with a certain frequency and amplitude, which are determined by the coefficients before the sine and cosine terms. This periodicity is essential when the curve forms a repeating structure, like a helix.

Helix Description

A helix is a three-dimensional spiral structure, characterized by a constant radius from a central axis and a uniform pitch, which is the height of one complete turn of the helix. Helices are seen everywhere in nature, from the coiling of a spring to the structure of DNA.

The curve described in the exercise turns out to represent a helix due to the intertwined sine and cosine functions in its vector components. However, this particular helix is non-uniform, meaning that its radius and pitch vary over its length which is determined by the coefficients in front of each trigonometric function. By modifying these coefficients, we alter the tightness and diameter of the spiral. The helix's shape in the exercise is dictated by the changing values of \(t\), making it expand and contract as it winds itself around a central point or axis. Its three-dimensional path offers a classic example of how trigonometry and vector components work together to create complex and fascinating shapes.