Vector Algebra
Cross product of unit vectors
Grade 12
Question:
<p>If \(\hat{u}\) and \(\hat{v}\) are unit vectors and \(\theta\) is the acute angle between them, then \(2\hat{u}\times 3\hat{v}\) is a unit vector for</p>
<p>exactly two values of \(\theta\).</p>
<p>more than two values of \(\theta\).</p>
<p>no value of \(\theta\).</p>
<p>exactly one value of \(\theta\).</p>
Step-by-Step Solution
Key Concept: The magnitude of a cross product of unit vectors depends on the sine of the angle between them: |a⃗ × b⃗| = |a⃗||b⃗|sin(θ). For 2û × 3v̂ to be a unit vector, we need 2·3·sin(θ) = 1, giving sin(θ) = 1/6.
Step 1: Recall that for vectors a⃗ and b⃗, the magnitude of their cross product is |a⃗ × b⃗| = |a⃗||b⃗|sin(θ), where θ is the angle between them. Step 2: Given û and v̂ are unit vectors: |û| = |v̂| = 1. We need |2û × 3v̂| = 1. Step 3: Calculate: |2û × 3v̂| = 2·3·|û||v̂|sin(θ) = 6·1·1·sin(θ) = 6sin(θ) Step 4: For this to equal 1 (unit vector): 6sin(θ) = 1, so sin(θ) = 1/6. Step 5: Since θ is acute and sin(θ) = 1/6, we have θ = sin⁻^1(1/6). ∴ Answer: D
Correct Answer: D