उत्तर (38)

समाधान

$\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k}$

$\overrightarrow{b}=-\hat{i}-8 \hat{j}+2 \hat{k}$

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

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

$(\vec{b}-\vec{c}) \times \vec{a}=0$

$\overrightarrow{b}-\overrightarrow{c}=\lambda \vec{\alpha}$

$\overrightarrow{b}=\overrightarrow{c}+\lambda \vec{\alpha}$

$-\hat{i}-8 \hat{j}+2 k=\left(4 \hat{i}+c _2 \hat{j}+c _3 k\right)+\lambda(\hat{i}+\hat{j}+k)$

$\lambda+4=-1 \Rightarrow \lambda=-5$

$\lambda+c _2=-8 \Rightarrow c _2=-3$

$\lambda+c _3=2 \Rightarrow c _3=7$

$\overrightarrow{c}=4 \hat{i}-3 \hat{j}+7 k$

$\cos \theta=\dfrac{12-12+7}{\sqrt{26} \cdot \sqrt{74}}=\dfrac{7}{\sqrt{26} \cdot \sqrt{74}}=\dfrac{7}{2 \sqrt{481}}$

$\tan ^{2} \theta=\dfrac{625 \times 3}{49}$

$\left[\tan ^{2} \theta\right]=38$