Problem 14 - Calculus Early Transcendentals

Question

\(7-14\) Sketch the curve with the given vector equation. Indicate with an arrow the direction in which \(t\) increases. $$ \mathbf{r}(t)=\cos t \mathbf{i}-\cos t \mathbf{j}+\sin t \mathbf{k} $$

Step-by-Step Solution

Answer

The curve is a helix moving upwards as \( t \) increases.

Step 1: Understand the Vector Equation

The given vector equation is \( \mathbf{r}(t) = \cos t \mathbf{i} - \cos t \mathbf{j} + \sin t \mathbf{k} \). This represents a curve in three-dimensional space. The curve is parametrized by \( t \), meaning as \( t \) changes, the position on the curve changes.

Step 2: Identify the Components of the Vector

The vector equation can be broken down into three components: \( x(t) = \cos t \), \( y(t) = -\cos t \), and \( z(t) = \sin t \). These equations define how the \( x \), \( y \), and \( z \) coordinates change as a function of \( t \).

Step 3: Explore the Trace of the Curve

By examining \( x(t) = \cos t \) and \( y(t) = -\cos t \), we find that \( y = -x \). This identifies that in the \( xy \)-plane, the curve is a line that lies on \( y = -x \). When \( z(t) = \sin t \) is considered, it suggests that as \( t \) varies, the curve traces a path that extends upward and downward in the \( z \)-direction.

Step 4: Project the Path into the XYZ Space

In the 3D space, \( x^2 + y^2 = \cos^2 t + (-\cos t)^2 = \cos^2 t + \cos^2 t = 2\cos^2 t \). Since \( z^2 = \sin^2 t \), the cosines and sines suggest a periodic repetition, where the trajectory of the path varies with \( t \).

Step 5: Sketch the Curve

In a 3D coordinate system, draw the XYZ axes. Plot points using some selected values like \( t = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \). For example, at \( t=0 \), \( \mathbf{r}(0) = \mathbf{i} - \mathbf{j} \) and at \( t=\frac{\pi}{2} \), \( \mathbf{r}(\frac{\pi}{2}) = \mathbf{k} \), progressing through selected \( t \) values, more of the path is visible. Connect these points smoothly to visualize the curve.

Step 6: Indicate the Direction of Increasing t

For an increase in \( t \), the path moves from one plotted point to the next along the curve. Use an arrow to show this direction, marking a direction that corresponds with increasing \( t \) values.

Key Concepts

3D Coordinate Systems

A 3D coordinate system is like a grid for three-dimensional space. It allows us to locate points using three numbers, known as coordinates. These coordinates are labeled as \( x \), \( y \), and \( z \), each representing a dimension in space. Imagine the 3D system as an extension of the flat 2D graph with an added dimension for depth.

• The \( x \)-axis is the horizontal line, similar to a ruler lying flat on a table.
• The \( y \)-axis is the vertical line, like a flagpole standing upright.
• The \( z \)-axis extends forwards and backwards, like a path through a tunnel.

All three axes meet at a point called the origin. In 3D graphs, each point's position is determined by how far along it is on each axis. For example, a point \( (2, -1, 3) \), tells us that:

• It's 2 units along the \( x \)-axis,
• -1 unit along the \( y \)-axis, and
• 3 units along the \( z \)-axis.

Parametric Equations

Parametric equations are a way to express the coordinates of points on a curve by using a parameter \( t \). It's like telling a story of how a point moves along the curve over time. Here, the position of the point is defined by separate equations for \( x \), \( y \), and \( z \). As \( t \) changes, these equations show how the point moves in 3D space. Let's break down how this works:- Each component of the vector equation \( \mathbf{r}(t) \) is a function of \( t \).- Considering \( x(t) = \cos t \), \( y(t) = -\cos t \), and \( z(t) = \sin t \), we can see how the point travels across the \( x \), \( y \), and \( z \) directions.The parameter \( t \), often represents time, guiding us to plot different points along the curve based on its value. This method is particularly handy for sketching complex paths, which aren't easily expressed in one formula.

XYZ Components

The \( xyz \) components are the building blocks of every point in the 3D coordinate system. Each component acts like a coordinate telling part of the point's full story in space. These are the "legs" which a point stands on in the imaginary 3D world.

• \( x \)-component: This is like the left-right movement, controlled by the \( x(t) = \cos t \) in our example.
• \( y \)-component: This manages the front-back movement, with the value \( y(t) = -\cos t \).
• \( z \)-component: Normally dealing with up-down movements, here it's \( z(t) = \sin t \).

These individual equations allow us to visualize how a point moves in the 3D world, piecing together a full journey from simple directional movements. As \( t \) increases or decreases, the combination of \( x, y, \) and \( z \) shifting gives rise to the trajectory of the curve.