Find the shortest distance between the lines- r- - - - 2- - 3k- - t2- - 3- - 4k- and r- - 2- - 4- - 5k- - s3- - 4- - 5k-

Equation of given lines: r⃗ = î + 2ĵ + 3k̂ + t(2î + 3ĵ + 4k̂) and r⃗ = 2î + 4ĵ + 5k̂ + s(3î + 4ĵ + 5k̂) ∴a⃗_⃗1⃗ = î + 2ĵ + 3k̂, b⃗_⃗1⃗ = 2î + 3ĵ + ...

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Byju's Answer

Standard XII

Mathematics

Shortest Distance between Two Skew Lines

Find the shor...

Question

Find the shortest distance between the lines,
$\to r =^i + 2^j + 3^k + t (2^i + 3^j + 4^k)$ and $\to r = 2^i + 4^j + 5^k + s (3^i + 4^j + 5^k)$

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Solution

Equation of given lines:
$\to r =^i + 2^j + 3^k + t (2^i + 3^j + 4^k)$
and $\to r = 2^i + 4^j + 5^k + s (3^i + 4^j + 5^k)$
$∴_{1} =^i + 2^j + 3^k,_{1} = 2^i + 3^j + 4^k$
and $_{2} = 2^i + 4^j + 5^k,_{2} = 3^i + 4^j + 5^k$
$∴_{2} -_{1} = (2^i + 4^j + 5^k) - (^i + 2^j + 3^k)$
$\Rightarrow_{2} -_{1} =^i + 2^j + 2^k$
$_{1} \times_{2} = ∣ ∣ ∣ ∣ ∣ \begin{matrix} ^i & ^j & ^k 2 & 3 & 4 3 & 4 & 5 \end{matrix} ∣ ∣ ∣ ∣ ∣$
$=^i (15 - 16) -^j (10 - 12) +^k (8 - 9)$
$=^i + 2^j -^k$
$∴ |_{1} \times_{2} | = | -^i + 2^j -^k |$
$= \sqrt{(- 1)^{2} + 2^{2} + (- 1)^{2}}$
$= \sqrt{1 + 4 + 1} = \sqrt{6}$
$∴$ shortest distance
$= \frac{(_{2} -_{1}) (_{1} \times_{2})}{|_{1} \times_{2} |}$
$= \frac{(^i + 2^j + 2^k) . (-^i + 2^j -^k)}{\sqrt{6}}$
$= \frac{- 1 + 4 - 2}{\sqrt{6}} = \frac{1}{\sqrt{6}}$ .

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Find the shortest distance between the lines,→r=^i+2^j+3^k+t(2^i+3^j+4^k) and →r=2^i+4^j+5^k+s(3^i+4^j+5^k)

Find the shortest distance between the lines,
$\to r =^i + 2^j + 3^k + t (2^i + 3^j + 4^k)$ and $\to r = 2^i + 4^j + 5^k + s (3^i + 4^j + 5^k)$