If the shortest distance between the line joining the points (1, 2, 3) and (2, 3, 4), and the line x - 1 / 2 = y + 1 / - 1 = z - 2 / 0 is alpha, then 28alpha^2 is equal to ____________.

Points (1,2,3) and (2,3,4) L_1: ((x-1) / 1)=((y-2) / 1)=((2-3) / 1) L_2: (x-1 / 2)=(y+1 / -1)=(z-2 / 0) vec b_1=hat i+hat j+hat k vec b_2=2 hat i-hat j+0 hat k ->a_1-->a_2=0 hat i-3 hat j-hat k \ d=|frac(bara_1-bara_2) * (n_1 x n_2)|n_1 x n_2|| \ =|frac6-3sqrt(9+1+4)|=frac3sqrt(14)=alpha \ 28 alpha^2=(28 x 9 / 14)=18

If the shortest distance between the line joining the p... | null (Mathematics)

Points

(1,2,3)

and

(2,3,4)

L_{1}: \frac{(x-1)}{1}=\frac{(y-2)}{1}=\frac{(2-3)}{1}

L_{2}: \frac{x-1}{2}=\frac{y+1}{-1}=\frac{z-2}{0}

\vec{b}_{1}=\hat{i}+\hat{j}+\hat{k}

\vec{b}_{2}=2 \hat{i}-\hat{j}+0 \hat{k}

\begin{aligned} & \overrightarrow{a_{1}}-\overrightarrow{a_{2}}=0 \hat{i}-3 \hat{j}-\hat{k} \\\\ & d=\left|\frac{\left(\bar{a}_{1}-\bar{a}_{2}\right) \cdot\left(n_{1} \times n_{2}\right)}{\left|n_{1} \times n_{2}\right|}\right| \\\\ &=\left|\frac{6-3}{\sqrt{9+1+4}}\right|=\frac{3}{\sqrt{14}}=\alpha \\\\ & 28 \alpha^{2}=\frac{28 \times 9}{14}=18 \end{aligned}

Explanation

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