bar{a}, bar{b}, bar{c} are three unit vectors | vedclass

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(A) Given that $\bar{a}, \bar{b}, \bar{c}$ are unit vectors with angles $\frac{\pi}{3}$ between them. The scalar triple product $[\bar{a} \bar{b} \bar

Vedclass · Accepted Answer

$\frac{3}{4}$

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(A) Given that $\bar{a}, \bar{b}, \bar{c}$ are unit vectors with angles $\frac{\pi}{3}$ between them. The scalar triple product $[\bar{a} \bar{b} \bar{c}] = |\bar{a}| |\bar{b} 	imes \bar{c}| \cos(\frac{\pi}{6}) = 1 \cdot \sin(\frac{\pi}{3}) \cdot \sin(\frac{\pi}{3}) \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2} = \frac{3\sqrt{3}}{8}$.Let $\bar{u} = \bar{b} 	imes \bar{c}$, $\bar{v} = \bar{c} 	imes \bar{a}$, $\bar{w} = \bar{a} 	imes \bar{b}$.Taking the dot product of the given equation $x \bar{a} + y \bar{b} + z \bar{c} = p \bar{u} + q \bar{v} + r \bar{w}$ with $\bar{a}, \bar{b}, \bar{c}$ respectively, we use the reciprocal system properties.Summing the components, we find that the ratio $\frac{x+y+z}{p+q+r}$ simplifies based on the symmetry of the vectors.Given the symmetry, $x=y=z$ and $p=q=r$.Using the projection properties, the ratio evaluates to $\frac{3}{4}$.

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