Question 1302d

Vectors II: Lines and Planes
Perpendicular Distance between Lines

Question

The lines l1{l_1} and l2{l_2} are given by
l1:r=(−433)+λ(−24−5),  λ∈R,l2:r=(−543)+μ(−24−5),  μ∈R.\begin{align*} &l_1: \mathbf{r} = \begin{pmatrix} - 4 \\ 3 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} - 2 \\ 4 \\ - 5 \end{pmatrix}, \; \lambda \in \mathbb{R}, \\ &l_2: \mathbf{r} = \begin{pmatrix} - 5 \\ 4 \\ 3 \end{pmatrix} + \mu \begin{pmatrix} - 2 \\ 4 \\ - 5 \end{pmatrix}, \; \mu \in \mathbb{R}. \end{align*}
Find the perpendicular distance between the two lines.

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