Vector Algebra
Vector Rotation and Complex Representation
Grade 12
Question:
<p>For some non-zero vector <strong>V</strong>, if the sum of <strong>V</strong> and the vector obtained from <strong>V</strong> by rotating it by ∠2α equals to the vector obtained from <strong>V</strong> by rotating it by ∠α, then the value of α, is</p>
<p>(a) \(2n\pi ± \frac{\pi}{3}\)</p>
<p>(b) \(n\pi ± \frac{\pi}{3}\)</p>
<p>(c) \(2n\pi ± \frac{2\pi}{3}\)</p>
<p>(d) \(n\pi ± \frac{2\pi}{3}\)</p>
Step-by-Step Solution
Key Concept: Use complex number representation of rotated vectors to convert the geometric condition into an algebraic equation.
Let V be rotated by angle θ. Representing vectors in complex form or using components: V + V (rotated by 2α) = V (rotated by α). In complex notation: z + ze^{i2α} = ze^{iα}, where z represents V . Dividing by z: 1 + e^{i2α} = e^{iα}. Using e^{iθ} = cos(θ) + i·sin(θ): 1 + cos(2α) + i·sin(2α) = cos(α) + i·sin(α). Comparing: 1 + cos(2α) = cos(α) and sin(2α) = sin(α). Using 1 + cos(2α) = 2cos^2(α): 2cos^2(α) = cos(α), giving cos(α)(2cos(α) − 1) = 0. Thus cos(α) = 0 or cos(α) = 1/2. This yields α = 2nπ ± π/3.
Correct Answer: A