If A(1,−2,−1), B(4,0,−3), C(1,2,−1), D(2,−4,−5) , find the distance between AB and CD.

Engineering

Mathematics

Vector Equation of a Line

Shortest Distance Between Two Lines

Distance from a Plane

Question

If A(1,−2,−1), B(4,0,−3), C(1,2,−1), D(2,−4,−5) , find the distance between AB and CD.

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Equation of

\vec{AB} = \frac{x - 1}{3} = \frac{y + 2}{2} = \frac{z + 1}{- 2}

\vec{CD} = \frac{x - 1}{1} = \frac{y - 2}{- 6} = \frac{z + 1}{- 4}

Distance between them is given by

∣ \begin{array}{ccc} x_{1} - x_{2} & y_{1} - y_{2} & z_{1} - z_{2} \\ ℓ_{1} & m_{1} & n_{1} \\ ℓ_{2} & m_{2} & n_{2} \end{array} ∣

= ∣ \begin{array}{ccc} 0 & 4 & 0 \\ 3 & 2 & - 2 \\ 1 & - 6 & - 4 \end{array} ∣

= (–4)(–12 + 2) = 40.