Problem 10 - Essential Calculus Early Transcendentals

Question

\(5-12=\) Sketch the curve with the given vector equation. Indicate with an arrow the direction in which \(t\) increases. $$ \mathbf{r}(t)=t^{2} \mathbf{i}+t \mathbf{j}+2 \mathbf{k} $$

Step-by-Step Solution

Answer

The curve is a 3D parabola on the plane \( z=2 \) with \( t \) increasing from left to right.

Step 1: Identify the components of the vector equation

The vector equation is given as \( \mathbf{r}(t) = t^2 \mathbf{i} + t \mathbf{j} + 2 \mathbf{k} \). Here, \( t^2 \) is the component in the \( x \)-direction, \( t \) in the \( y \)-direction, and the constant 2 in the \( z \)-direction.

Step 2: Understand the path described by the vector equation

The path of the vector \( \mathbf{r}(t) \) is determined by the components given. As \( t \) changes, \( x = t^2 \), \( y = t \), and \( z = 2 \). This describes a curve in 3D space.

Step 3: Find points on the curve

To sketch the curve, find several points by substituting different \( t \) values. For example, when \( t = 0 \), \( \mathbf{r}(0) = 0 \mathbf{i} + 0 \mathbf{j} + 2 \mathbf{k} = (0, 0, 2) \). When \( t = 1 \), \( \mathbf{r}(1) = 1 \mathbf{i} + 1 \mathbf{j} + 2 \mathbf{k} = (1, 1, 2) \). And when \( t = -1 \), \( \mathbf{r}(-1) = 1 \mathbf{i} - 1 \mathbf{j} + 2 \mathbf{k} = (1, -1, 2) \).

Step 4: Sketch the curve

Plot the points found in the previous step: \((0, 0, 2)\), \((1, 1, 2)\), and \((1, -1, 2)\) on a 3D coordinate system. The curve lies on a plane where \( z = 2 \), and it will resemble a parabola opening along the \( x \)-axis, with \( y \) increasing proportionally to \( t \).

Step 5: Indicate the direction of increasing \( t \)

To show the direction in which \( t \) increases, draw an arrow along the curve from left to right based on the order of the plotted points: you move from \((0, 0, 2)\) to \((1, 1, 2)\) as \( t \) increases. This indicates the direction of the curve as \( t \) increases.

Key Concepts

Vector Equations

Vector equations play a crucial role in describing spaces and movements in mathematics and physics. With vector equations, you can specify the path of an object or any point that moves through space.
Understanding vector equations involves recognizing that these equations are comprised of different direction components. For the given vector equation, \(\mathbf{r}(t) = t^2 \mathbf{i} + t \mathbf{j} + 2 \mathbf{k}\), you have:

• \(t^2 \mathbf{i}\) impacting the \(x\)-axis,
• \(t \mathbf{j}\) affecting the \(y\)-axis,
• \(2 \mathbf{k}\) being constant along the \(z\)-axis.

This means as the parameter \(t\) changes, each component of \(\mathbf{r}(t)\) alters the position in its respective direction.
A firm grasp of vector components lets you predict the movement of the vector path in 3D space, thus making sketching, analyzing, and understanding these paths intuitive and practical.

3D Coordinate Systems

A 3D coordinate system allows us to visualize and plot points in three-dimensional space using three mutually perpendicular axes: the \(x\)-axis, \(y\)-axis, and \(z\)-axis.
In the exercise provided, the curve is plotted within this type of space. Each point on this curve corresponds to a specific location given by the vector function \(\mathbf{r}(t)\). Examples of points include:

• When \(t = 0\), the point is \((0, 0, 2)\).
• When \(t = 1\), the point shifts forward to \((1, 1, 2)\).
• When \(t = -1\), the point is \((1, -1, 2)\).

The plane where this curve lies is parallel to the \(xy\)-plane. But it's interesting to note it is offset by 2 units up along the \(z\)-axis.
Understanding the interaction between these axes and the movement along them for any parameter \(t\) helps tremendously in visualizing curves and other elements within a 3D coordinate system effectively.

Parametric Curves

Parametric curves are defined by parameters that let us easily describe curves with individual components linked directly to the parameter values.
For the vector equation \(\mathbf{r}(t) = t^2 \mathbf{i} + t \mathbf{j} + 2 \mathbf{k}\), the parameter \(t\) determines your position on the curve. As \(t\) varies, you get a corresponding point on the curve. These points change because:

• The \(x\)-coordinate is always \(t^2\), shaping a parabola.
• The \(y\)-coordinate increases linearly with \(t\).
• The \(z\)-coordinate stays constant at 2.

This form of representation is powerful as it represents motion and changes over time or another parameter.
By sketching parametric curves, you observe how varying parameters affect their shapes and paths, which is fundamentally different from functions defined by explicit equations in terms of \(x\), \(y\), and \(z\) only.