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Chapter 14: Problem 10

Sketch and represent parametrically. Segment from \(1+i\) to \(4-2 i\)

Short Answer

Expert verified
Parametric form: \(z(t) = (1 + 3t) + i(1 - 3t)\) for \(t \in [0, 1]\).

Step by step solution

01

Understand the Points

The segment starts at the point \(1+i\) and ends at the point \(4-2i\). In the complex plane, \(1+i\) translates to the Cartesian point \((1, 1)\), and \(4-2i\) translates to the Cartesian point \((4, -2)\).
02

Calculate the Direction Vector

To find the direction vector of the segment, subtract the coordinates of the starting point from the ending point: \((4 - 1) + (-2 - 1)i = 3 - 3i\).So, the direction vector is \((3, -3)\).
03

Parametrize the Segment

The parametric equation of a line segment can be written as:\[ z(t) = z_0 + t \cdot (z_1 - z_0) \]where \( z_0 = 1+i\) is the starting complex number and \(z_1 - z_0 = 3 - 3i\) is the direction vector, and \(t\) ranges from 0 to 1. Substituting the values:\( z(t) = (1+i) + t(3-3i) = (1 + 3t) + i(1 - 3t)\) for \(t \in [0, 1]\).
04

Sketch the Segment

To sketch the segment, plot the points \((1, 1)\) and \((4, -2)\) on the complex plane. Draw a straight line connecting these two points. This line represents the segment from the point \(1+i\) to \(4-2i\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complex Numbers
Complex numbers are numbers that have both a real part and an imaginary part. They are usually denoted as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. The "\(i\)" in the imaginary part represents the square root of \(-1\). This notation allows complex numbers to be visualized as points in a two-dimensional plane.

For example:
  • The complex number \(1+i\) is represented by the point \((1, 1)\) in the plane.
  • Similarly, \(4-2i\) is depicted as the point \((4, -2)\).
These representations provide a direct link between complex numbers and geometry. They allow us to explore relationships, like the segment between two points as given in the original exercise.
Direction Vectors
A direction vector provides information about the direction and magnitude of a line segment. It's calculated by subtracting the coordinates of the starting point from the ending point.

For the segment from \(1+i\) to \(4-2i\), the direction vector is found by:
  • Subtracting real parts: \(4 - 1 = 3\)
  • Subtracting imaginary parts: \(-2 - 1 = -3\)
Thus, the direction vector is \((3,-3)\) or \(3-3i\) in complex number form. This tells us that if you start at \(1+i\), moving \(3\) units right and \(-3\) units down will take you to \(4-2i\). The vector is essential for understanding how to move from one point to another, giving both direction and length.
Complex Plane
The complex plane is a graphical representation of complex numbers similar to the Cartesian coordinate system. It consists of a horizontal axis for the real part and a vertical axis for the imaginary part.

When plotting a complex number:
  • The real part is the x-coordinate.
  • The imaginary part is the y-coordinate.
For example, to plot the number \(1+i\), you move to \((1, 1)\) on the plane. The line segment we are focusing on, from \(1+i\) to \(4-2i\), appears as a straight line connecting the points \((1,1)\) and \((4,-2)\) on this plane. The complex plane allows for a straightforward visual interpretation of complex numbers and operations on them, such as addition, subtraction, and parametrization.
Parametrization
Parametrization is a way of expressing a line, curve, or geometric object using a parameter, often "\(t\)", that varies over a certain interval. For the segment from \(1+i\) to \(4-2i\), the parametric equation is:

\[ z(t) = (1+i) + t(3-3i) = (1 + 3t) + i(1 - 3t) \] where \(t\) ranges from \(0\) to \(1\).

This equation reflects how the position changes as \(t\) goes from \(0\) (start) to \(1\) (end):
  • At \(t=0\), \(z(0) = 1+i\), marking the start of the segment.
  • At \(t=1\), \(z(1) = 4-2i\), marking the end of the segment.
This method highlights how lines or curves can be represented in terms of motion, which is key to understanding more complex mathematical models and systems.

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Most popular questions from this chapter

Evaluate (showing the details and using partial fractions if necessary) \(\oint_{c} \frac{d z}{z^{2}+1}, c:(a)|z|=\frac{1}{2}, \quad(b)|z-1|=\frac{a}{2}\) (counterclockwise)

Find end sketch the path and its orientation given by: $$z(t)=1+2 t+8 t^{2}(-1 \leq t \leq 1)$$

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Sketch and represent parametrically. Segment from \(a+i b\) to \(c+i d\)

Sketch and represent parametrically. \(|z-2+3 i|=4\) (counterclockwise)
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