Question

Let \(z_1\) and \(z_2\) be two complex numbers, then the dot product of \(z_1\) and \(z_2\) is :

• A. \(|z_1||z_2|\)
• B. \(|z_1||z_2|\sin\theta\)
• C. \(|z_1||z_2|\cos\theta\)
• D. \(|z_1||z_2|\tan\theta\)

Hint:

Remember that the dot product of any two vectors (or complex numbers viewed as vectors) is always associated with the cosine of the angle between them:
\[ \mathbf{u} \cdot \mathbf{v} = \|\mathbf{u}\| \|\mathbf{v}\| \cos\theta \]
This rule is universal across geometry and vector algebra.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Complex numbers can be represented as 2D vectors in the Argand plane.
If \(z_1 = x_1 + i y_1\) and \(z_2 = x_2 + i y_2\), their corresponding vector representations are \(\mathbf{v_1} = (x_1, y_1)\) and \(\mathbf{v_2} = (x_2, y_2)\).
The dot product (scalar product) of these two vectors is defined in terms of their magnitudes and the angle \(\theta\) between them.
Key Formula or Approach:
The dot product of two vectors \(\mathbf{v_1}\) and \(\mathbf{v_2}\) is given by:
\[ \mathbf{v_1} \cdot \mathbf{v_2} = |\mathbf{v_1}| |\mathbf{v_2}| \cos\theta \]
In complex number notation:
- \(|\mathbf{v_1}| = |z_1|\)
- \(|\mathbf{v_2}| = |z_2|\)

Step 2: Detailed Explanation:
Let us represent the complex numbers in polar form:
\[ z_1 = |z_1|(\cos\theta_1 + i\sin\theta_1) \]
\[ z_2 = |z_2|(\cos\theta_2 + i\sin\theta_2) \]
Their vector components are:
\[ \mathbf{v_1} = (|z_1|\cos\theta_1, |z_1|\sin\theta_1) \]
\[ \mathbf{v_2} = (|z_2|\cos\theta_2, |z_2|\sin\theta_2) \]
Now, calculate the dot product using components:
\[ \mathbf{v_1} \cdot \mathbf{v_2} = (|z_1|\cos\theta_1)(|z_2|\cos\theta_2) + (|z_1|\sin\theta_1)(|z_2|\sin\theta_2) \]
Factor out the product of the moduli:
\[ \mathbf{v_1} \cdot \mathbf{v_2} = |z_1||z_2| (\cos\theta_1\cos\theta_2 + \sin\theta_1\sin\theta_2) \]
Using the trigonometric identity \(\cos(A - B) = \cos A\cos B + \sin A\sin B\):
\[ \mathbf{v_1} \cdot \mathbf{v_2} = |z_1||z_2| \cos(\theta_1 - \theta_2) \]
Let \(\theta = \theta_1 - \theta_2\) be the angle between the two complex numbers.
Then:
\[ z_1 \cdot z_2 = |z_1||z_2| \cos\theta \]
This matches Option (C).

Step 3: Final Answer:
The correct option is (C).