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Towards Biot-Savart Law

Let e1 and ...

Question

Let $e_{1}$ and $e_{2}$ be unit vectors containing angle $θ$ . If $\frac{1}{2} | e_{1} - e_{2} | = sin λ θ$ , then find $λ$ .

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Solution

Given $e_{1}$ and $e_{2}$ are unit vector
$\Rightarrow | e_{1} | = 1 = | e_{2} |$
By parallelogram law of vector addition
$| \to a + \to b | = \sqrt{| a |^{2} + | b |^{2} + 2 | d b | cos θ}$
$| e_{1} - e_{2} | = \sqrt{| e_{1} |^{2} + | e_{2} |^{2} + 2 | e_{1} | | e_{2} | cos (180 - θ)}$
$| e_{1} - e_{2} | = \sqrt{1 + 1 - 2 cos θ}$
$| e_{1} - e_{2} | = \sqrt{2} \sqrt{1 - cos θ}$
$| e_{1} - e_{2} | = \sqrt{2} \sqrt{2 {sin}^{2} θ / 2}$
$| e_{1} - e_{2} | = \sqrt{2} \sqrt{2} sin θ / 2$
$| e_{1} - e_{2} | = 2 sin θ / 2$
$\frac{1}{2} | e_{1} - e_{2} | = sin θ / 2$ .............. $(1)$
Comparing $(1)$ with $\frac{1}{2} | e_{1} - e_{2} | = sin θ$
we get $λ = \frac{1}{2}$

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