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Multiplication with Vectors

Find the shor...

Question

Find the shortest distance between the following pair of lines.
$¯ r = (^i + 2^j +^k) + λ (2^i -^j + 3^k)$
$¯ r = (^i - 2^j +^k) + μ (3^i + 2^j - 5^k)$

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Solution

Given two lines, $_{1} = (^i + 2^j +^k) + λ (2^i -^j + 3^k)$
$_{2} = (^i - 2^j +^k) + μ (3^i + 2^j - 5^k)$
The shortest distance between $r_{1}$ and $r_{2}$ must be perpendicular to both
$∴$ Distance is along $(2^i -^j + 3^k) \times (3^i + 2^j - 5^k)$
$∣ ∣ ∣ ∣ \begin{matrix} i & j & k 2 & - 1 & 3 3 & 2 & - 5 \end{matrix} ∣ ∣ ∣ ∣ = 0 i + 19 j + 1 k$
$= 19 j + 7 k$
$∴$ Shortest distance $= | [(i + 2 j + k) - (i - 3 j - k)] (19 j + 7 k) |$
$= | (5 j + 2 k) (19 j + 7 k) |$
$= 109$

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