Sketch The Plane Curve With The Given Vector Equation. (b) Findr(t). (c) Sketch The Position Vectorr(t)and

Sketch The Plane Curve With The Given Vector Equation. (b) Find r(t). (c) Sketch The Position Vector r(t) and

Understanding the behavior of plane curves defined by vector equations is fundamental in vector calculus and differential geometry. These curves describe the motion of a point in the plane as a function of a parameter, often time \( t \). This article explores the process of sketching such a curve, deriving the position vector \( r(t) \), and visualizing the motion graphically. The discussion is structured into clear sections to aid comprehension and application of these concepts.

Introduction to Vector Equations of Plane Curves

What Is a Vector Equation?

A vector equation of a plane curve expresses the position of a point on the curve as a function of a parameter \( t \). Typically, it takes the form: \[ r(t) = x(t)\mathbf{i} + y(t)\mathbf{j} \] where:

• \( x(t) \) and \( y(t) \) are scalar functions of \( t \),
• \( \mathbf{i} \) and \( \mathbf{j} \) are the unit vectors in the x- and y-directions, respectively.

This compact formulation encapsulates both the x- and y-coordinates of the point at any given \( t \).

Common Types of Plane Curves

Plane curves can be classified based on their equations:

• Parametric curves: defined explicitly via \( r(t) \),
• Cartesian curves: expressed as a relation \( y = f(x) \),
• Polar curves: expressed as \( r = r(\theta) \).

In this discussion, we focus on parametric (vector) equations, which are most flexible for representing complex motions.

Sketching the Plane Curve From a Given Vector Equation

Step 1: Understand the Given Vector Equation

Suppose the vector equation is: \[ r(t) = x(t)\mathbf{i} + y(t)\mathbf{j} \] For example, consider: \[ r(t) = (a \cos t) \mathbf{i} + (b \sin t) \mathbf{j} \] where \( a \) and \( b \) are constants.

This particular form resembles the parametric equations of an ellipse:
\[
x(t) = a \cos t,\quad y(t) = b \sin t
\]

Step 2: Find the Domain of \( t \)

Determine the range of \( t \) over which the curve is traced:

• Typically, for trigonometric functions, \( t \) varies over \( [0, 2\pi] \),
• For other functions, analyze the behavior to identify the interval.

Step 3: Analyze the Behavior of the Coordinates

In the example:

• \( x(t) = a \cos t \) oscillates between \( -a \) and \( a \),
• \( y(t) = b \sin t \) oscillates between \( -b \) and \( b \).

Plotting these points for various \( t \) values illustrates the shape: an ellipse centered at the origin.

Step 4: Find the Cartesian Equation of the Curve

Eliminate the parameter \( t \) to find a Cartesian relation: \[ x = a \cos t \Rightarrow \cos t = \frac{x}{a} \] \[ y = b \sin t \Rightarrow \sin t = \frac{y}{b} \] Using the Pythagorean identity: \[ \sin^2 t + \cos^2 t = 1 \] substitute: \[ \left(\frac{y}{b}\right)^2 + \left(\frac{x}{a}\right)^2 = 1 \] This is the equation of an ellipse: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]

Step 5: Sketch the Curve

• Plot key points, such as \( (a, 0), (-a, 0), (0, b), (0, -b) \),
• Connect these points smoothly to form the ellipse,
• Use symmetry to aid in sketching.

Deriving the Position Vector \( r(t) \)

Step 1: Confirm the Parametric Functions

Given the vector equation, identify \( x(t) \) and \( y(t) \). For example: \[ r(t) = (a \cos t)\mathbf{i} + (b \sin t)\mathbf{j} \] So, \[ r(t) = \begin{bmatrix} a \cos t \\ b \sin t \end{bmatrix} \]

Step 2: Write the Position Vector Explicitly

Express the position vector explicitly: \[ r(t) = \langle x(t), y(t) \rangle \] which in the example is: \[ r(t) = \langle a \cos t,\, b \sin t \rangle \]

Step 3: Analyze the Range of \( r(t) \)

• The magnitude of \( r(t) \):

\[ |r(t)| = \sqrt{(a \cos t)^2 + (b \sin t)^2} \]

• The vector traces the ellipse in the plane as \( t \) varies over its domain.

Step 4: Determine the Velocity and Acceleration Vectors

• Velocity:

\[ v(t) = r'(t) = \frac{d}{dt} \langle a \cos t,\, b \sin t \rangle = \langle -a \sin t,\, b \cos t \rangle \]

• Acceleration:

\[ a(t) = r''(t) = \langle -a \cos t,\, -b \sin t \rangle \]

These derivatives are useful for understanding the motion's dynamics along the curve.

Sketching the Position Vector \( r(t) \) and the Curve

Visualizing the Curve

• Plot the points corresponding to various \( t \) values,
• Use parametric plots to draw the smooth path,
• Indicate the direction of motion with arrows, if desired.

Visualizing the Position Vector

• Draw the position vector from the origin to the point \( r(t) \),
• For selected \( t \)-values, illustrate the vector to demonstrate how the point moves over time,
• Highlight the initial position \( r(t_0) \) and subsequent positions to show the trajectory.

Additional Elements to Enhance the Sketch

• Show the axes with appropriate scales,
• Mark key points such as vertices and intercepts,
• Include tangent lines at specific points to illustrate the direction of motion,
• Use different colors to distinguish between the curve and the position vectors.

Summary and Practical Tips

Summary of the Process

To sketch a plane curve from a vector equation:

1. Identify \( x(t) \) and \( y(t) \),
2. Determine the domain of \( t \),
3. Find the Cartesian form by eliminating \( t \),
4. Plot key points and symmetry,
5. Draw the curve smoothly,
6. Visualize the position vector at various points,
7. Analyze the behavior of \( r(t) \), \( v(t) \), and \( a(t) \).

Practical Tips for Accurate Sketching

• Use a table of \( t \) values to find corresponding points,
• Pay attention to symmetry and periodicity,
• Sketch the curve lightly, then refine,
• Include direction arrows for better understanding of motion,
• Use software tools like graphing calculators or computer algebra systems for complex curves.

Conclusion

Mastering the process of sketching plane curves from vector equations enhances understanding of parametric motion and geometric properties. By systematically analyzing the functions \( x(t) \) and \( y(t) \), deriving the Cartesian form, and visualizing the position vector \( r(t) \), one gains a comprehensive view of the curve's shape and the dynamics along it. Whether dealing with simple circles or complex ellipses, these techniques form the foundation for advanced studies in vector calculus, physics, and engineering applications.