QJEEM24Maafe4: JEE Main 2024 Physics - Class 11 JEEM24Maafe4 Solution: Let and be three vectors such that . If the angle between the…

Question

Verified Solution • 2025–26 Session moderate

Let

\overrightarrow{\mathrm{a}}=\hat{i}+\hat{j}+\hat{k}, \overrightarrow{\mathrm{b}}=-\hat{i}-8 \hat{j}+2 \hat{k}

and

\overrightarrow{\mathrm{c}}=4 \hat{i}+\mathrm{c}_2 \hat{j}+\mathrm{c}_3 \hat{k}

be three vectors such that

\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{a}}=\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}}

. If the angle between the vector

\overrightarrow{\mathrm{c}}

and the vector

3 \hat{i}+4 \hat{j}+\hat{k}

is

\theta

, then the greatest integer less than or equal to

\tan ^2 \theta

is _______________.

Accepted Answer

\begin{aligned} &amp; \vec{a}=\hat{i}+\hat{j}+k \\\\ &amp; \vec{b}=\hat{i}+8 \hat{j}+2 k \\\\ &amp; \vec{c}=4 \hat{i}+c_2 \hat{j}+c_3 k \\\\ &amp; \vec{b} \times \vec{a}=\vec{c} \times \vec{a} \\\\ &amp; (\vec{b}-\vec{c}) \times \vec{a}=0 \\\\ &amp; \vec{b}-\vec{c}=\lambda \vec{\alpha} \\\\ &amp; \vec{b}=\vec{c}+\lambda \vec{\alpha}\end{aligned}

\begin{aligned} &amp; -\hat{\mathrm{i}}-8 \hat{\mathrm{j}}+2 \mathrm{k}=\left(4 \hat{\mathrm{i}}+\mathrm{c}_2 \hat{\mathrm{j}}+\mathrm{c}_3 \mathrm{k}\right)+\lambda(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\mathrm{k}) \\\\ &amp; \lambda+4=-1 \Rightarrow \lambda=-5 \\\\ &amp; \lambda+\mathrm{c}_2=-8 \Rightarrow \mathrm{c}_2=-3 \\\\ &amp; \lambda+\mathrm{c}_3=2 \Rightarrow \mathrm{c}_3=7 \\\\ &amp; \overrightarrow{\mathrm{c}}=4 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}+7 \mathrm{k}\end{aligned}

\begin{aligned} &amp; \cos \theta=\frac{12-12+7}{\sqrt{26} \cdot \sqrt{74}}=\frac{7}{\sqrt{26} \cdot \sqrt{74}}=\frac{7}{2 \sqrt{481}} \\\\ &amp; \tan ^2 \theta=\frac{625 \times 3}{49} \\\\ &amp; {\left[\tan ^2 \theta\right]=38}\end{aligned}

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