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Find cos θ for ā = 2î + 3ĵ - 4k̂ and b̄ = î + 2ĵ + 3k̂ (where θ is the angle between them):

A1
B-4/√406
C4/√406
D0
Answer & Solution
Correct answer: B. -4/√406
ā · b̄ = (2)(1) + (3)(2) + (-4)(3) = 2 + 6 − 12 = −4. |ā| = √(4+9+16) = √29. |b̄| = √(1+4+9) = √14. cos θ = (ā · b̄) / (|ā||b̄|) = −4 / √(29 × 14) = −4 / √406. Negative cos → obtuse angle (θ ≈ 101.5°).
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