उत्तर: (3501)

समाधान:

सूत्र: वेक्टर के अदिश त्रिगुणित उत्पाद , दो वेक्टर के अदिश उत्पाद

$[\vec{u} \vec{v} w]=\vec{u} \cdot(\vec{v} \times \vec{w})$

$\min .(|u||\vec{v} \times \vec{w}| \cos \theta)=-\alpha \sqrt{3401}$

$\Rightarrow \cos \theta=-1$

$|u|=\alpha$ (दिया गया)

$|\vec{v} \times \vec{w}|=\sqrt{3401}$

$\vec{v} \times \vec{w}= \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \alpha & 2 & -3 \\ 2 \alpha & 1 & -1\end{vmatrix} $

$\vec{v} \times \vec{w}=\hat{i}-5 \alpha \hat{j}-3 \alpha \hat{k}$

$|\vec{v} \times \vec{w}|=\sqrt{1+25 \alpha^{2}+9 \alpha^{2}}=\sqrt{3401}$

$34 \alpha^{2}=3400$

$\alpha^{2}=100$

$\alpha=10$

(क्योंकि $\alpha>0$ )

इसलिए $\quad \vec{u}=\lambda(\hat{i}-5 \alpha \hat{j}-3 \alpha \hat{k})$

$\vec{u}=\sqrt{\lambda^{2}+25 \alpha^{2} \lambda^{2}+9 \alpha^{2} \lambda}$

$\alpha^{2}=\lambda^{2}(1+25 \alpha^{2}+9 \alpha^{2})$

$100=\lambda^{2}(1+34 \times 100) $

$\Rightarrow \lambda^{2}= \dfrac{100}{3401} = \dfrac{m}{n}$

$\vec{u} \cdot \hat{i} = (\lambda \hat{i} - 5 \alpha \lambda \hat{j} - 3 \lambda \alpha \hat{k}) \cdot \hat{i}$

$\vec{u} \cdot \hat{i} = \lambda$

$|\vec{u} \cdot \hat{i}|^2 = \lambda^2 = \dfrac{100}{3401} = \dfrac{m}{n}$

$\therefore m+n = 100+3401 = 3501$