Question

There is a one-to-one correspondence between two-dimensional vectors and complex numbers. Show that the real and imaginary parts of the product \(z_{1} z_{2}^{*}\) (the star denotes comples conjugate) are respectively the scalar product and \(\pm\) the magnitude of the vector product of the vectors corresponding to \(z_{1}\) and \(z_{2}\).

Short answer

The real part of \(z_1 z_2^*\) is the scalar product, and the imaginary part is the magnitude of the vector product of the vectors.

Step-by-step solution

- Understand the problem

The exercise asks to show that the real and imaginary parts of a product of two complex numbers, where one is conjugated, correspond to the scalar product and the magnitude of the vector product respectively of their vector representations.

- Represent Complex Numbers as Vectors

Let complex numbers be represented as two-dimensional vectors: \( z_1 = a + bi \rightarrow \textbf{v}_1 = (a, b) \) \( z_2 = c + di \rightarrow \textbf{v}_2 = (c, d) \)

- Define the Conjugate

The complex conjugate of \( z_2 \) is \( z_2^* = c - di \).

- Multiply Complex Numbers

Compute the product \( z_1 \times z_2^* \): \[ z_1 \times z_2^* = (a + bi)(c - di) \]Using distributive property:\[ z_1 \times z_2^* = ac - adi + bci - bdi^2 \]Since \( i^2 = -1 \):\[ z_1 \times z_2^* = ac + bd + (bc - ad)i \]

- Identify Real and Imaginary Parts

From the product, we see the real part is \( ac + bd \) and the imaginary part is \( bc - ad \).

- Relate to Scalar Product

The scalar (dot) product of vectors \( \textbf{v}_1 = (a, b) \) and \( \textbf{v}_2 = (c, d) \) is defined as: \[ \textbf{v}_1 \bullet \textbf{v}_2 = ac + bd \]This matches the real part of \( z_1 \times z_2^* \).

- Relate to Vector Product Magnitude

The magnitude of the vector (cross) product in two dimensions is given by: \[ |\textbf{v}_1 \times \textbf{v}_2| = |ad - bc| \]This matches the imaginary part of \( z_1 \times z_2^* \) up to a sign, confirming it is \( \boldsymbol{\text{±}} |ad - bc| \).

Key concepts

Scalar Product

The scalar product, also known as the dot product, is an operation that takes two vectors and returns a single number. To find the scalar product, you multiply the corresponding components of each vector and then add the results. For two vectors \(\textbf{v}_1 = (a, b)\) and \(\textbf{v}_2 = (c, d)\), the scalar product is calculated as:
\[ \textbf{v}_1 \bullet \textbf{v}_2 = ac + bd \]
This operation is important because it relates directly to the real part of the complex number product in our exercise. When we multiply \(z_1\) and the conjugate of \(z_2\), the real part of the product aligns with the scalar product of their respective vector representations.

Vector Product Magnitude

The vector product, commonly referred to as the cross product, is another way to multiply two vectors, but it results in a vector perpendicular to the originals in three dimensions.
However, in two dimensions, we often refer to its magnitude. For vectors \(\textbf{v}_1 = (a, b)\) and \(\textbf{v}_2 = (c, d)\), the magnitude of the vector product is given by:
\[ |\textbf{v}_1 \times \textbf{v}_2| = |ad - bc| \]
This value appears as the magnitude of the imaginary part when two complex numbers are multiplied, with one being conjugated. This confirms that the imaginary part of the product corresponds to \(\pm |ad - bc| \).

Complex Conjugate

The complex conjugate of a complex number is straightforward to find. For a complex number \(z = c + di\), its conjugate (denoted as \(z^*\)) is:
\[ z^* = c - di \]

• The real part remains the same.
• The imaginary part's sign is reversed.

Understanding the concept of conjugation is essential for solving our exercise since multiplying a complex number with the conjugate of another complex number yields a result where the real and imaginary parts have meaningful vector interpretations.

Dot Product

The dot product is another term for the scalar product, and it serves as a measure of how much two vectors point in the same direction. Given vectors \(\textbf{v}_1 = (a, b)\) and \(\textbf{v}_2 = (c, d)\), the dot product is:
\[ \textbf{v}_1 \bullet \textbf{v}_2 = ac + bd \]
This is significant because the real part of the product of \(z_1\) and the conjugate of \(z_2\) is exactly this dot product. Understanding dot product helps interpret results from vector and complex number manipulations.

Cross Product

The cross product results in a vector that is perpendicular to the original two vectors in three dimensions.
However, in our two-dimensional case, we consider the magnitude. For \(\textbf{v}_1 = (a, b)\) and \(\textbf{v}_2 = (c, d)\), it's:
\[ |\textbf{v}_1 \times \textbf{v}_2| = |ad - bc| \]
This matches the imaginary part of the product of \(z_1\) and conjugated \(z_2\). Understanding this helps us make sense of complex number multiplication in context, linking it back to essential vector operations.