<p>If a unit vector \(\vec{a}\) makes angles \(\pi/3\) with \(\hat{i}\), \(\pi/4\) with \(\hat{j}\) and \(\theta \in (0, \pi)\) with \(\hat{k}\), then a value of \(\theta\) is ______ (in degrees).</p>
Step-by-Step Solution
Key Concept: For any unit vector, the sum of squares of direction cosines equals 1: cos²α + cos²β + cos²γ = 1. This constraint uniquely determines the third angle when two angles are given.
Step 1: For a unit vector a making angles π/3, π/4, and θ with i , j , k respectively, the direction cosines are cos(π/3), cos(π/4), and cos(θ). Step 2: Apply the fundamental property: cos^2(π/3) + cos^2(π/4) + cos^2(θ) = 1 Step 3: Substitute values: (1/2)^2 + (1/√2)^2 + cos^2(θ) = 1 Therefore: 1/4 + 1/2 + cos^2(θ) = 1 3/4 + cos^2(θ) = 1 cos^2(θ) = 1/4 Step 4: Taking square root: cos(θ) = ±1/2 Step 5: For θ ∈ (0, π): • If cos(θ) = 1/2, then θ = π/3 = 60° • If cos(θ) = -1/2, then θ = 2π/3 = 120° ∴ Answer: 60 or 120 (both values are valid)
Correct Answer: 60