Find the shortest distance between the lines r- - - - 2- - -k- r- -2- - -k- - 3- -5- -2k-

The shortest distance between the lines of the form r⃗ = x _ 1 î + y _ 1 ĵ + z _ 1 k̂ +λ ( l _ 1 î + m _ 1 ĵ + n _ 1 k̂ ) and r⃗ = x _ 2 î + y _ 2 ĵ + z ...

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Byju's Answer

Standard XII

Mathematics

Shortest Distance between Two Skew Lines

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Question

Find the shortest distance between the lines
$\to r =^i +^j + λ (2^i -^j +^k)$
$\to r = 2^i +^j -^k + μ (3^i - 5^j + 2^k)$

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Solution

The shortest distance between the lines of the form $\to r = x_{1}^i + y_{1}^j + z_{1}^k + λ (l_{1}^i + m_{1}^j + n_{1}^k) a n d \to r = x_{2}^i + y_{2}^j + z_{2}^k + λ (l_{2}^i + m_{2}^j + n_{2}^k)$ is given by
$∣ ∣ ∣ ∣ \begin{matrix} x_{1} - x_{2} & y_{1} - y_{2} & z_{1} - z_{2} l_{1} & m_{1} & n_{1} l_{2} & m_{2} & n_{2} \end{matrix} ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ \begin{matrix} - 1 & 0 & 1 2 & - 1 & 1 3 & - 5 & 2 \end{matrix} ∣ ∣ ∣ ∣ = | - 10 | = 10 u n i t s$

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Similar questions

Q. Find the shortest distance between the lines

l_{1}

and

l_{2}

whose vector equation are given by

\to r =^i +^j + λ (2^i -^j +^k)

\to r = 2^i +^j -^k + μ (3^i - 5^j + 2^k)

Q. Find the shortest distance between line

l_{1}

and

l_{2}

whose vector equations are

\to r =^i +^j + μ (2^i -^j +^k)

................ (i)
and

\to r = 2^i +^i -^k + μ (3^i - 5^j + 2^k)

............. (ii)

Q. Find the shortest distance between the lines

\to r = (^i + 2^j +^k) + λ (^i -^j +^k)

and

\to r = 2^i -^j -^k + μ (2^i +^j + 2^k)

Q. Find the shortest distance between the lines.

\to r =^i + 2^j +^k + λ (^i -^j +^k)

\to r = 2^i -^j -^k + μ (2^i +^j + 2^k)

Q. Find the shortest distance between the following pairs of parallel lines.

\to r = (^i +^j) + λ (2^i +^j -^k)

and

\to r = (^i -^j +^k) + μ (-^i +^j -^k)

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Find the shortest distance between the lines→r=^i+^j+λ(2^i−^j+^k)→r=2^i+^j−^k+μ(3^i−5^j+2^k)

The shortest distance between the lines of the form →r=x1^i+y1^j+z1^k+λ(l1^i+m1^j+n1^k)and→r=x2^i+y2^j+z2^k+λ(l2^i+m2^j+n2^k) is given by∣∣ ∣∣x1−x2y1−y2z1−z2l1m1n1l2m2n2∣∣ ∣∣=∣∣ ∣∣−1012−113−52∣∣ ∣∣=|−10|=10units

Find the shortest distance between the lines
$\to r =^i +^j + λ (2^i -^j +^k)$
$\to r = 2^i +^j -^k + μ (3^i - 5^j + 2^k)$