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Question:

Two vectors A and B are defined as A = a^i and B = a(cosωt^i + sinωt^j), where a is a constant and ω = π/6 rads⁻¹. If ||A + B|| = √3||A - B|| at time t = τ for the first time, then the value of τ, in seconds, is _______.

Solution:

L.H.S. = ||A + B|| = 2acos(ωt/2)
R.H.S. = √3||A - B|| = √3 × 2asin(ωt/2)
L.H.S. = R.H.S. ⇒ 2acos(ωt/2) = √3(2asin(ωt/2))
⇒ tan(ωt/2) = 1/√3
⇒ ωt/2 = π/6
⇒ ωt = π/3
⇒ (π/6)t = π/3
t = 2.00 sec