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Magazine Chapter 14: Problem 6
Find end sketch the path and its orientation given by: $$z(n)=3+4 t+5 e^{i t}(\pi \leq t \leq 2 \pi)$$
Short Answer
Expert verified
The path is a counter-clockwise circle of radius 5, with the center moving from \(3+4\pi\) to \(3+8\pi\).
Step by step solution
01
Identify the Components of the Function
The function given is a complex function in the form of \(z(t) = 3 + 4t + 5e^{it}\). Here, \(t\) is a parameter that varies from \(\pi\) to \(2\pi\). The constant terms \(3\) and \(4t\) represent a linear movement in the real part, while \(5e^{it}\) represents a circular path in the complex plane.
02
Understand the Circular Motion Component
The term \(5e^{it}\) defines a path that traces a circle in the complex plane. This is because \(e^{it}\) maps to points on the unit circle in complex numbers, and multiplying by 5 scales this to a circle of radius 5. Therefore, \(5e^{it}\) traces a circle with center at the origin and radius 5.
03
Combine Linear and Circular Movements
The function \(z(t) = 3 + 4t + 5e^{it}\) combines a linear motion \(3 + 4t\) with the circular motion \(5e^{it}\). The term \(3 + 4t\) indicates the center of the circle moves along a straight line from \((3+4\pi, 0)\) to \((3+8\pi, 0)\) as \(t\) goes from \(\pi\) to \(2\pi\).
04
Determine Start and End Points
We calculate the start point by plugging \(t = \pi\): \[ z(\pi) = 3 + 4\pi + 5e^{i\pi} = 3 + 4\pi - 5 \left(= (-2 + 4\pi) \right) \]The end point when \(t = 2\pi\): \[ z(2\pi) = 3 + 8\pi + 5e^{i2\pi} = 3 + 8\pi + 5
ight(= (8 + 8\pi) \right) \]Thus, the path begins at \((-2 + 4\pi)\) and ends at \((8 + 8\pi)\).
05
Sketch the Path
Sketch a line for the movement of the circle鈥檚 center from \(3 + 4\pi\) to \(3 + 8\pi\). Then draw part of a circle at each new center point from \((-2 + 4\pi)\) to \((8 + 8\pi)\) with radius 5. The circle will be completed as \(t\) moves from \(\pi\) to \(2\pi\), tracing counter-clockwise due to the \(e^{it}\) term.
06
Determine the Orientation
Since \(e^{it}\) traces the unit circle counter-clockwise as \(t\) increases, the orientation of the path is counter-clockwise. The full path completes one loop in the time from \(\pi\) to \(2\pi\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Complex Plane
Imagine you鈥檙e standing on a flat, endless sheet of paper that represents the complex plane. This plane is critical in understanding complex functions, where every point can be described by a complex number, such as \( z = x + yi \). Here, \(x\) is the real part, and \(yi\) is the imaginary part. Think of the complex plane as a two-dimensional graph:
- Real Axis: This is like the familiar x-axis on a standard graph, depicting real number values.
- Imaginary Axis: Imagine a new dimension at a right angle to the real axis; this represents the imaginary component.
Circular Motion
Circular motion in the context of complex functions can be fascinating. Consider the term \(5e^{it}\) from our example. This represents a circle traced out in the complex plane. Here鈥檚 how it works:
- Unit Circle: The expression \(e^{it}\) defines a unit circle with a radius of 1, as it cycles through different angles \(t\) around the circle.
- Scaling by 5: Multiply by 5, and you鈥檙e expanding this to a new circle with a radius of 5, centered at the origin.
- Counter-Clockwise Movement: As \(t\) increases from \(\pi\) to \(2\pi\), the motion is counter-clockwise, completing the circular path.
Parametric Equations
Parametric equations can describe dynamic paths, including those in the complex plane. They are like storytelling in mathematics. For complex numbers, a parametric equation involves one or multiple expressions that define a path as a parameter, typically \( t \), changes:
- Linear Component: \(3 + 4t\) illustrates a linear path in our example, showing movement along the real axis as \(t\) increases.
- Circular Component: \(5e^{it}\) generates the rotating path around a circle, modulating the overall motion.
- Combined Movement: The complete expression, \(z(t) = 3 + 4t + 5e^{it}\), blends linear and circular movements, giving rise to more complex trajectories.
