Question

Create a vector-valued function whose graph matches the given description. A vertically oriented helix with radius of 2 that starts at (2,0,0) and ends at \((2,0,4 \pi)\) after 1 revolution on \([0,2 \pi]\).

Short answer

The vector-valued function is \(\mathbf{r}(t) = \langle 2\cos(t), 2\sin(t), 2t \rangle\).

Step-by-step solution

Understand the Helix Parameters

A helix is defined by its radius, vertical stretch, and orientation. Here, the radius is given as 2, the vertical change (or height) per revolution is \(4\pi\), and the helix is aligned along the z-axis. This suggests a vertically oriented helix in a cylindrical coordinate form \((r \cos(t), r \sin(t), z)\) where \(z = \frac{4\pi}{2\pi}t = 2t\).

Determine the Helix Components

Given the radius is 2, the components of the function in terms of parameter \(t\) are:\[ x(t) = 2 \cos(t) \quad \text{and} \quad y(t) = 2 \sin(t) \quad \text{where} \ t \in [0, 2\pi].\]The vertical component \(z(t)\) increases linearly from 0 to \(4\pi\) as \(t\) moves from 0 to \(2\pi\), so \(z(t) = 2t.\)

Construct the Vector-Valued Function

Using the components determined in Step 2, the vector-valued function for the helix is:\[\mathbf{r}(t) = \langle 2\cos(t), 2\sin(t), 2t \rangle\]for \(t\) in the interval \([0, 2\pi]\).

Verify the Endpoints

Verify the starting and ending points by substituting \(t = 0\) and \(t = 2\pi\) into the function. When \(t = 0\), \(\mathbf{r}(0) = \langle 2, 0, 0 \rangle\), which is the starting point (2,0,0). When \(t = 2\pi\), \(\mathbf{r}(2\pi) = \langle 2, 0, 4\pi \rangle\), which is the ending point (2,0,4\pi). Both match the given description.

Key concepts

Helix

A helix is a fascinating curve that spirals around an axis in three dimensions, like a spring or the threads on a screw. In our case, this helix is vertically oriented, meaning it wraps around the z-axis. The main characteristics defining this helix include its radius, which determines how far it spirals away from the axis, and its vertical stretch, which indicates how much it ascends along the z-axis per revolution. This particular helix from the problem has a radius of 2, and it completes one full turn while ascending from the point (2,0,0) to the point (2,0,4π). As such, the helix we are considering spirals with points dropping onto a circle of radius 2 in the xy-plane at each layer, while independently moving vertically upwards. It is often visualized in a cylindrical coordinate system due to its easy translation into such a system involving rotational symmetry.

Parametric Equations

The representation of the helix using parametric equations helps us to define its path as a function of a single parameter, usually denoted as \(t\). This parameter varies over an interval, in the case of our helix, from 0 to \(2\pi\), which maps directly to a full circle. The parametric equations are based on what we infer from rotation and elevation components:

• \( x(t) = 2\cos(t) \)
• \( y(t) = 2\sin(t) \)
• \( z(t) = 2t \)

These equations emerge from recognizing that the circular motion in the xy-plane is related to the trigonometric functions cosine and sine, which ensure the periodic nature of rotation. The increasing linear term in \( z(t) = 2t \) indicates the steady upward movement per unit of \(t\), allowing it to achieve the final height of \(4\pi\) for one complete revolution.

Cylindrical Coordinates

Cylindrical coordinates are particularly convenient for dealing with objects or functions that exhibit rotational symmetry around a vertical axis, much like this helix. They extend polar coordinates by adding a height value, making them suitable for three-dimensional analysis where one axis (in this case, z-axis) aligns with the object's orientation.
In cylindrical coordinates, any point is described using

• the radial distance \(r\) from the origin,
• an angle \(\theta\) with the positive x-axis, and
• height \(z\).

For our helical path:- The radius \(r\) remains consistent at 2.- The angle \(\theta = t\) simply tracks the curves in the horizontal plane.- The height \(z = 2t\) describes vertical climbing.
By comprehending how the helix transitions from this coordinate system to the familiar Cartesian x-y system through equations, students can visualize each layer's horizontal circle and its position along the z-axis, offering deeper insight into how such spirals are composed in space.