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The line $l_{1}$ passes through the point $(1, 2, - 1)$ and the line $l_{2}$ passes through $(2, - 1, 3) .$ The line $\frac{x - 1}{- 1} = \frac{y + α}{2} = \frac{z + β}{- 2}$ intersects $l_{1}$ and $l_{2}$ at right angles at A and B respectively for some $α, β \in R .$ The minimum value of AB is

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a

1

b

$\frac{13}{3}$

c

$\frac{1}{3}$

d

5

answer is D.

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Detailed Solution

${\vec{a}}_{1} = (1, 2, - 1), {\vec{a}}_{2} = (2, - 1, 3), {\vec{b}}_{1} \times {\vec{b}}_{2} = (- 1, 2, - 2)$

$∴$ minimum value of $A B = | \frac{(\vec{a_{2}} - \vec{a_{1}}) . \vec{b_{1}} \times \vec{b_{2}}}{| \vec{b} \times \vec{b_{2}} |} |$

$= \frac{15}{3} = 5$

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The line l1 passes through the point (1, 2, −1) and the line l2 passes through (2,−1,3). The line x−1−1=y+α2=z+β−2 intersects l1 and l2 at right angles at A and B respectively for some α,β∈ R. The minimum value of AB is