Two vectors and are defined as and , where is a constant and . If at time for the first time, the value of , in seconds, is ______. [2018]

Question

Two vectors $\vec{A}$ and $\vec{B}$ are defined as $\vec{A} = a \hat{i}$ and $\vec{B} = a (\cos ω t \hat{i} + \sin ω t \hat{j})$ , where $a$ is a constant and $ω = \frac{π}{6} {rad s}^{- 1}$ . If $| \vec{A} + \vec{B} |$ $= \sqrt{3}$ $| \vec{A} - \vec{B} |$ at time $t = τ$ for the first time, the value of $τ$ , in seconds, is ______. [2018]

Smart Achievers · Accepted Answer

(2)

$Given:$ $| \vec{A} + \vec{B} |$ $= \sqrt{3}$ $| \vec{A} - \vec{B} |$

$∴$ $| a \hat{i} + a \cos ω t \hat{i} + a \sin ω t \hat{j} |$ $= \sqrt{3}$ $| a \hat{i} - a \cos ω t \hat{i} - a \sin ω t \hat{j} |$

$\Rightarrow$ $| (1 + \cos ω t) \hat{i} + \sin ω t \hat{j} |$ $= \sqrt{3}$ $| (1 - \cos ω t) \hat{i} - \sin ω t \hat{j} |$

$\sqrt{2 + 2 \cos ω t} = \sqrt{3} \sqrt{2 - 2 \cos ω t}$

$∴$ $1 + \cos ω t = 3 (1 - \cos ω t)$

$\Rightarrow 4 \cos ω t = 2$ $∴ \cos ω t = \frac{1}{2}$ $or, ω t = \frac{π}{3}$

$∴$ $\frac{π}{6} \times τ = \frac{π}{3}$ $∴ τ = 2.00 seconds$

Solution

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Solution

Q. Two vectors A→ and B→ are defined as A→=ai^ and B→=a(cosωt i^+sinωt j^), where a is a constant and ω=π6 rad s-1. If |A→+B→|=3|A→-B→| at time t=τ for the first time, the value of τ, in seconds, is ______. [2018]

Ans. (2) Given: |A→+B→|=3|A→-B→| ∴|ai^+acosωt i^+asinωt j^|=3|ai^-acosωt i^-asinωt j^| ⇒|(1+cosωt)i^+sinωt j^|=3|(1-cosωt)i^-sinωt j^| 2+2cosωt=32-2cosωt ∴ 1+cosωt=3(1-cosωt) ⇒4cosωt=2 ∴ cosωt=12 or, ωt=π3 ∴ π6×τ=π3 ∴ τ=2.00 seconds

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