Vector Algebra
Scalar triple product and dot product
Grade None
Question:
<p>Let <strong>d</strong> be a non-zero vector such that <strong>d</strong>·<strong>a</strong> = [<strong>a</strong><strong>b</strong><strong>c</strong>]cos y = −<strong>d</strong>·(<strong>b</strong>+<strong>c</strong>). If sin x + cos y + 2 = 0, then the minimum value of x² + y² is:</p>
<p>A) π²/4</p>
<p>B) 5π²/4</p>
<p>C) π²</p>
<p>D) 3π²/4</p>
Step-by-Step Solution
Key Concept: Use the constraint d·a = [abc]cos y = −d·(b+c) to establish a relationship between vectors, then apply the given trigonometric constraint sin x + cos y + 2 = 0 to find the minimum of x² + y².
Step 1: Analyze the constraint sin x + cos y + 2 = 0, which gives sin x + cos y = −2. Step 2: Since −1 ≤ sin x ≤ 1 and −1 ≤ cos y ≤ 1, the only way their sum equals −2 is if sin x = −1 AND cos y = −1 simultaneously. Step 3: From sin x = −1, we get x = −π/2 + 2πk (or x = 3π/2 + 2πk for k ∈ ℤ). The principal value is x = −π/2. Step 4: From cos y = −1, we get y = π + 2πm for m ∈ ℤ. The principal value is y = π. Step 5: Calculate x^2 + y^2 = (−π/2)^2 + π^2 = π^2/4 + π^2 = 5π^2/4. ∴ Answer: B
Correct Answer: B