QJEEM23Ma7041: JEE Main 2023 Physics - Class 11 JEEM23Ma7041 Solution: Let be three vectors such that and . If the angle between and…

Question

Verified Solution • 2025–26 Session moderate

Let

\vec{a}, \vec{b}, \vec{c}

be three vectors such that

|\vec{a}|=\sqrt{31}, 4|\vec{b}|=|\vec{c}|=2

and

2(\vec{a} \times \vec{b})=3(\vec{c} \times \vec{a})

.

If the angle between

\vec{b}

and

\vec{c}

is

\frac{2 \pi}{3}

, then

\left(\frac{\vec{a} \times \vec{c}}{\vec{a} \cdot \vec{b}}\right)^{2}

is equal to __________.

Accepted Answer

2(\vec{a} \times \vec{b})=3(\vec{c} \times \vec{a})

\vec{a} \times(2 \vec{b}+3 \vec{c})=0

\begin{aligned}&amp; \vec{a}=\lambda(2 \vec{b}+3 \vec{c}) \\\\&amp; |\vec{a}|^{2}=\lambda^{2}\left(4|b|^{2}+9|c|^{2}+12 \vec{b} \cdot \vec{c}\right) \\\\&amp; 31=31 \lambda^{2} \\\\&amp; \lambda=\pm 1 \\\\&amp; \vec{a}=\pm(2 \vec{b}+3 \vec{c}) \\\\&amp; \frac{|\vec{a} \times \vec{c}|}{|\vec{a} \cdot \vec{b}|}=\frac{2|\vec{b} \times \vec{c}|}{2 \vec{b} \cdot \vec{b}+3 \vec{c} \cdot \vec{b}} \\\\&amp; |\vec{b} \times \vec{c}|^{2}=\frac{1}{4} \cdot 4-\left(1-\frac{1}{2}\right)^{2} \\\\&amp; =\frac{3}{4} \\\\&amp; \therefore \frac{|\vec{a} \times \vec{c}|}{|\vec{a} \cdot \vec{b}|}=\frac{\sqrt{3}}{2 \cdot \frac{1}{4}-\frac{3}{2}}=\frac{\sqrt{3}}{-1} \\\\&amp; \left(\frac{|\vec{a} \times \vec{c}|}{|\vec{a} \cdot \vec{b}|}\right)^{2}=3\end{aligned}

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