Problem 2 - Advanced Engineering Mathematics

Question

Answer Problems 1-16 without referring back to the text. Fill in the blank or answer true/false. If \(z\) is a point in the third quadrant, then \(i z\) is in the _________ quadrant.

Step-by-Step Solution

Answer

\(iz\) is in the fourth quadrant.

Step 1: Determine the Location of z

If by assumption, the point \( z \) is located in the third quadrant of the complex plane, it means that its real part \( \text{Re}(z) \) is negative and its imaginary part \( \text{Im}(z) \) is also negative.

Step 2: Understand the Effect of Multiplying by i

Multiplying a complex number by \( i \) results in a 90-degree counterclockwise rotation in the complex plane. This means if \( z = a + bi \), then \( i z = i(a + bi) = -b + ai \).

Step 3: Compute the New Location for iz

From Step 2, since \( z = a + bi \) and \( a, b < 0 \) because \( z \) is in the third quadrant, then when we multiply \( z \) by \( i \), we get \( i z = -b + ai \). This means \( -b \) is positive and \( a \) is negative.

Step 4: Determine the Quadrant for i z

Since the real part \( -b > 0 \) is positive and the imaginary part \( a < 0 \) is negative, this places \( i z \) in the fourth quadrant of the complex plane.

Key Concepts

Quadrants of the Complex Plane

The complex plane is similar to the Cartesian plane, but it allows us to visualize complex numbers. It features two axes: the horizontal axis represents the real part of a complex number, and the vertical axis represents the imaginary part. The plane is divided into four quadrants:

• First Quadrant: Here, both the real part and the imaginary part are positive.
• Second Quadrant: The real part is negative, but the imaginary part is positive.
• Third Quadrant: Both the real and imaginary parts are negative.
• Fourth Quadrant: The real part is positive, while the imaginary part is negative.

This quadrant system helps determine the characteristics and operations on complex numbers, such as rotations and multiplications, and where the resulting complex number will lie on the plane.
Understanding the quadrants helps in determining the implications of mathematical operations on complex numbers.

Multiplication by Imaginary Unit

Multiplication by the imaginary unit, denoted by the symbol \( i \), is a unique operation in the complex plane. The imaginary unit \( i \) is defined by the property \( i^2 = -1 \). When a complex number \( z = a + bi \) is multiplied by \( i \), it results in a special operation known as rotation in the complex plane. Specifically:

• \( i(a + bi) = ai + i^2b = ai - b \)
• This means that the real and imaginary parts are transformed into \(-b\) and \(a\), respectively.

By applying this multiplication, the complex number undergoes a 90-degree counterclockwise rotation. This rotational property helps visualize how complex numbers shift their position when involved in various operations, providing a deeper geometric understanding of their behaviors in equations and transformations.

Complex Plane Rotation

Complex plane rotation is an intriguing concept that revolves around how complex numbers change position when rotated. The mechanism of rotation often involves multiplying by \( i \) or its powers.

When you multiply a complex number like \( z = a + bi \) by \( i \), the number undergoes a 90-degree rotation counterclockwise. This operation swaps and negates the parts in a specific manner:

• The real part \( a \) becomes the imaginary part \( ai \)
• The imaginary part \( bi \) becomes the negative real part \(-b\)

In our exercise, a number initially in the third quadrant ends up in the fourth quadrant due to this rotation. This understanding of rotation in the complex plane clarifies how complex numbers are transformed through multiplication by \( i \), showing its vital role in shifting and reorienting positions within these four quadrants.