Question

Sketch the vector-valued function on the given interval in \(\mathbb{R}^{3}\). Technology may be useful in creating the sketch. $$ \vec{r}(t)=\langle 3 \cos t, \sin t, t / \pi\rangle \text { on }[0,2 \pi] $$

Short answer

The vector function describes a helical path rising from 0 to 2 in R³.

Step-by-step solution

Analyze the Components

Identify the components of the vector-valued function \(\vec{r}(t)=\langle 3 \cos t, \sin t, t/\pi\rangle\). These components represent the coordinates: \(x(t) = 3 \cos t\), \(y(t) = \sin t\), and \(z(t) = t/\pi\). We will sketch how these change as \(t\) varies from \(0\) to \(2\pi\).

Determine the Domain and Range

Since \(t\) ranges from \(0\) to \(2\pi\), the values \(x(t) = 3 \cos t\) shall oscillate between \(-3\) and \(3\), \(y(t) = \sin t\) will oscillate between \(-1\) and \(1\), and \(z(t) = t/\pi\) will range from \(0\) to \(2\).

Visualize the Path in 3D Space

Consider how these changes reflect a path in three-dimensional space. As \(t\) increases from \(0\) to \(2\pi\), the \((x, y)\) components form an elliptical path due to the relationship between the \(\cos\) and \(\sin\) functions, while the \(z\) component increases linearly from \(0\) to \(2\).

Recognize the Shape

The path formed by these parameters is a helical structure. The ellipse \((x(t), y(t)) = (3 \cos t, \sin t)\) spins around the \(z\)-axis, and the increase in \(z(t)\) simulates the vertical climb up the \(z\)-axis, creating a helix.

Create a Sketch

To sketch this in \(\mathbb{R}^{3}\), draw a circular base along the \(xy\)-plane where \(3 \cos t\) and values of \(\sin t\) dictate the horizontal ellipse's shape. Then visualize each complete loop of the helix rising steadily as \(z\) increases until it completes two paths from \(z = 0\) to \(z = 2\).

Key concepts

3D space visualization

To understand vector-valued functions like \(\vec{r}(t)=\langle 3 \cos t, \sin t, t / \pi\rangle\) in three-dimensional space, visualizing their paths helps tremendously. Picture a three-dimensional graph where each point on the graph represents a vector-head. It's like plotting multiple points to see where a line travels over time.

For this function, the values of \( x(t) = 3 \cos t \) and \( y(t) = \sin t \) create an elliptical trace on the \(xy\)-plane, like drawing with an Etch A Sketch. Meanwhile, as \(t\) progresses from \(0\) to \(2\pi\), the \(z(t) = t / \pi\) component elevates this ellipse upwards, extending into a third dimension.

In visualization:

• The \(x\)-axis, \(y\)-axis, and \(z\)-axis define the 3D coordinate space.
• The vector function's values describe a path or curve that seems to spiral upwards.
• An understanding of how the component functions interact helps form this three-dimensional path.

Using tools like graphing calculators or software that illustrate 3D plots can provide a real-time feeling of how the curve behaves.

helix

A helix is one of the most intriguing shapes in mathematics and science. Imagine a coil spring or the spiral staircase — that's the kind of shape we're dealing with in this exercise. When we say that the function \(\vec{r}(t)=\langle 3 \cos t, \sin t, t / \pi\rangle\) forms a helix, we're describing this spiraling, ascending path.

The function's elliptical path in the \(xy\)-plane mixed with the steady upward climb represented by \(z(t) = t / \pi\) makes this possible.

Key features of the helix in this vector-valued function:

• The elliptical shape caused by \(3 \cos t\) and \(\sin t\) means the path curves around a central axis, here, it's the \(z\)-axis.
• The \(z(t)\) component adds a vertical dimension, causing the spiral to move upwards rather than staying flat on the \(xy\)-plane.
• As \(t\) changes from \(0\) to \(2\pi\), two complete turns of the helix are traced, with each loop climbing higher.

Understanding helical structures involves seeing both the revolving movement in the \(xy\)-plane and the linear progression along the \(z\)-axis.

domain and range

Defining the domain and range of a vector-valued function is crucial for effectively understanding the complete picture of the path it describes. In our case, the domain of the function \(\vec{r}(t)=\langle 3 \cos t, \sin t, t / \pi\rangle\) is the closed interval \([0, 2\pi]\). This range for \(t\) gives us a full cycle of the function's motion and shape.

For each component:

• The range of \(x(t)=3 \cos t\) oscillates between \(-3\) and \(3\), due to the nature of the cosine function which ranges from \(-1\) to \(1\).
• Similarly, the range of \(y(t)=\sin t\) spans from \(-1\) to \(1\).
• For \(z(t)=t/\pi\), as \(t\) moves between \(0\) and \(2\pi\), \(z(t)\) ranges from \(0\) to \(2\).

Grasping these domains and ranges allows you to predict how the helix forms and limits where it goes within the 3D space. The entire path of the helix is encapsulated within these bounds, forming two complete spirals as \(t\) increases.