 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2024/paper-1/11/d/i/solution

Write

z_{j} = x_{j} + i y_{j}

Q = x_{1} x_{2} + y_{1} y_{2} + i (y_{1} x_{2} - x_{1} y_{2}) .

Therefore the real part is the dot product of the corresponding vectors:

ℜ Q = x_{1} x_{2} + y_{1} y_{2} = v_{1} \cdot v_{2} .

For nonzero vectors, this is

∣ z_{1} ∣∣ z_{2} ∣ cos (ar g z_{1} - ar g z_{2})

, measuring their alignment.

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