If bar{a}, bar{b}, bar{c}, bar{d} are unit ve | vedclass

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(C) Let $\bar{u} = \bar{a} 	imes \bar{b}$ and $\bar{v} = \bar{c} 	imes \bar{d}$.Given that $\bar{a}, \bar{b}, \bar{c}, \bar{d}$ are unit vectors,$|\

Vedclass · Accepted Answer

$\frac{3}{8}$

Vedclass · Answer

(C) Let $\bar{u} = \bar{a} 	imes \bar{b}$ and $\bar{v} = \bar{c} 	imes \bar{d}$.Given that $\bar{a}, \bar{b}, \bar{c}, \bar{d}$ are unit vectors,$|\bar{a}| = |\bar{b}| = |\bar{c}| = |\bar{d}| = 1$.Since $\bar{a} \cdot \bar{b} = \frac{1}{2}$,the angle between $\bar{a}$ and $\bar{b}$ is $\frac{\pi}{3}$,so $|\bar{a} 	imes \bar{b}| = |\bar{a}||\bar{b}| \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}$.Similarly,$|\bar{c} 	imes \bar{d}| = \frac{\sqrt{3}}{2}$.Using the vector triple product identity,$[\bar{a} \bar{b} \bar{d}] \bar{c} - [\bar{a} \bar{b} \bar{c}] \bar{d} = (\bar{a} 	imes \bar{b}) 	imes (\bar{c} 	imes \bar{d}) = \bar{u} 	imes \bar{v}$.The magnitude is $|\bar{u} 	imes \bar{v}| = |\bar{u}| |\bar{v}| \sin(	heta)$,where $	heta = \frac{\pi}{6}$.$|\bar{u} 	imes \bar{v}| = (\frac{\sqrt{3}}{2}) (\frac{\sqrt{3}}{2}) \sin(\frac{\pi}{6}) = \frac{3}{4} 	imes \frac{1}{2} = \frac{3}{8}$.

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