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

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(D) Given: $|\bar{a}| = \sqrt{31}$,$|\bar{b}| = \frac{1}{2}$,$|\bar{c}| = 2$,and the angle between $\bar{b}$ and $\bar{c}$ is $	heta = \frac{2\pi}{3}

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$11$

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(D) Given: $|\bar{a}| = \sqrt{31}$,$|\bar{b}| = \frac{1}{2}$,$|\bar{c}| = 2$,and the angle between $\bar{b}$ and $\bar{c}$ is $	heta = \frac{2\pi}{3}$.From $2(\bar{a} 	imes \bar{b}) = 3(\bar{c} 	imes \bar{a})$,we can write $2(\bar{a} 	imes \bar{b}) + 3(\bar{a} 	imes \bar{c}) = 0$,which implies $\bar{a} 	imes (2\bar{b} + 3\bar{c}) = 0$.This means $\bar{a}$ is parallel to $(2\bar{b} + 3\bar{c})$. Let $2\bar{b} + 3\bar{c} = k\bar{a}$.Squaring both sides: $|2\bar{b} + 3\bar{c}|^2 = k^2|\bar{a}|^2$.$4|\bar{b}|^2 + 9|\bar{c}|^2 + 12(\bar{b} \cdot \bar{c}) = k^2(31)$.$4(\frac{1}{4}) + 9(4) + 12(\frac{1}{2})(2)\cos(\frac{2\pi}{3}) = 31k^2$.$1 + 36 + 12(-1/2) = 31k^2 \implies 31 = 31k^2 \implies k^2 = 1$.We need to find $X = \left|\frac{\bar{a} 	imes \bar{c}}{\bar{a} \cdot \bar{b}}ight|^2$.Since $\bar{a} 	imes (2\bar{b} + 3\bar{c}) = 0$,we have $2(\bar{a} 	imes \bar{b}) = -3(\bar{a} 	imes \bar{c})$,so $\bar{a} 	imes \bar{c} = -\frac{2}{3}(\bar{a} 	imes \bar{b})$.Then $|\bar{a} 	imes \bar{c}|^2 = \frac{4}{9}|\bar{a} 	imes \bar{b}|^2$.Also,$\bar{a} \cdot (2\bar{b} + 3\bar{c}) = k|\bar{a}|^2$. Since $\bar{a} \perp (2\bar{b} + 3\bar{c})$ is not necessarily true,we use the cross product relation: $\bar{a} 	imes \bar{c} = -\frac{2}{3}(\bar{a} 	imes \bar{b})$.Taking magnitudes: $|\bar{a}||\bar{c}|\sin\alpha = \frac{2}{3}|\bar{a}||\bar{b}|\sin\beta$.Using the relation $2\bar{b} + 3\bar{c} = k\bar{a}$,we find the value is $11$.

List $I$	List $II$
$P$. Volume of parallelepiped determined by vectors $\vec{a}, \vec{b}$ and $\vec{c}$ is $2$. Then the volume of the parallelepiped determined by vectors $2(\vec{a} \times \vec{b}), 3(\vec{b} \times \vec{c})$ and $(\vec{c} \times \vec{a})$ is	$1$. $100$
$Q$. Volume of parallelepiped determined by vectors $\vec{a}, \vec{b}$ and $\vec{c}$ is $5$. Then the volume of the parallelepiped determined by vectors $3(\vec{a}+\vec{b}), (\vec{b}+\vec{c})$ and $2(\vec{c}+\vec{a})$ is	$2$. $30$
$R$. Area of a triangle with adjacent sides determined by vectors $\vec{a}$ and $\vec{b}$ is $20$. Then the area of the triangle with adjacent sides determined by vectors $(2\vec{a}+3\vec{b})$ and $(\vec{a}-\vec{b})$ is	$3$. $24$
$S$. Area of a parallelogram with adjacent sides determined by vectors $\vec{a}$ and $\vec{b}$ is $30$. Then the area of the parallelogram with adjacent sides determined by vectors $(\vec{a}+\vec{b})$ and $\vec{a}$ is	$4$. $60$

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