Vector Algebra
Angle Between Vectors
Grade 12
Question:
<p>Let \(\hat{a}\) and \(\hat{b}\) be two unit vectors. If the vectors \(\vec{c} = \hat{a} + 2\hat{b}\) and \(\vec{d} = 5\hat{a} - 4\hat{b}\) are perpendicular to each other, then the angle between \(\hat{a}\) and \(\hat{b}\) is</p>
<p>\(\dfrac{\pi}{6}\)</p>
<p>\(\dfrac{\pi}{2}\)</p>
<p>\(\dfrac{\pi}{3}\)</p>
<p>\(\dfrac{\pi}{4}\)</p>
Step-by-Step Solution
Key Concept: Two vectors are perpendicular if and only if their dot product equals zero. Use this condition along with the property that unit vectors have magnitude 1 to find the angle between them.
Step 1: Since c ⊥ d , we have c · d = 0. Step 2: Expand the dot product:
(â + 2b̂)·(5â - 4b̂) = 0
5(â·â) - 4(â·b̂) + 10(b̂·â) - 8(b̂·b̂) = 0 Step 3: Use |â| = |b̂| = 1, so â·â = 1 and b̂·b̂ = 1. Also â·b̂ = b̂·â:
5(1) - 4(â·b̂) + 10(â·b̂) - 8(1) = 0
5 + 6(â·b̂) - 8 = 0
6(â·b̂) = 3
â·b̂ = 1/2 Step 4: Since â·b̂ = |â||b̂|cosθ = (1)(1)cosθ = 1/2, we get cosθ = 1/2. Step 5: Therefore θ = 60° or π/3 radians. ∴ Answer: C
Correct Answer: C