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Shortest distance between two skew lines

This type of question, asked 8 times: 10 May 2023 S2, 9 May 2024 S2, 12 May 2024 S2, 21 Apr 2025 S1, 13 Apr 2026 S1, 19 Apr 2026 S1, 19 Apr 2026 S2, 20 Apr 2026 S1. All 8 questions below, oldest first.

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  1. Question 1

    10 May 2023 S2

    113
    The shortest distance between lines r⃗=(i^+2j^−k^)+λ(2i^+j^−3k^)\vec{r} = (\hat{i}+2\hat{j}-\hat{k}) + \lambda(2\hat{i}+\hat{j}-3\hat{k}) and r⃗=(2i^−j^+2k^)+μ(i^−j^+k^)\vec{r} = (2\hat{i}-\hat{j}+2\hat{k}) + \mu(\hat{i}-\hat{j}+\hat{k}) is

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  2. Question 2

    9 May 2024 S2

    121
    The shortest distance between the lines x−12=y−23=z−34\dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{4} and x−23=y−44=z−55\dfrac{x-2}{3}=\dfrac{y-4}{4}=\dfrac{z-5}{5} is

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  3. Question 3

    12 May 2024 S2

    103
    The shortest distance (in units) between the lines x+13=y+21=z+12\frac{x+1}{3}=\frac{y+2}{1}=\frac{z+1}{2} and r⃗=(2i^−2j^+3k^)+λ(i^+2j^)\vec{r}=(2\hat{i}-2\hat{j}+3\hat{k})+\lambda(\hat{i}+2\hat{j}) is

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  4. Question 4

    21 Apr 2025 S1

    104
    The shortest distance between the lines r‾=(4i^−j^)+λ(i^+2j^−3k^)\overline{r}= (4\widehat{i}-\widehat{j}) +\lambda(\widehat{i}+ 2\widehat{j}- 3\widehat{k}) and r‾=(i^−j^+2k^)+μ(2i^+4j^−5k^)\overline{r}= (\widehat{i}-\widehat{j}+ 2\widehat{k}) +\mu(2\widehat{i}+ 4\widehat{j}- 5\widehat{k}) is

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  5. Question 5

    13 Apr 2026 S1

    92
    The shortest distance between the lines, where the first line passes through (0,0,0)(0, 0, 0) and (2,0,3)(2, 0, 3) and the second line passes through (2,5,0)(2, 5, 0) and (0,4,0)(0, 4, 0) is

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  6. Question 6

    19 Apr 2026 S1

    95
    If pp is the shortest distance between the lines x+17=y+1−6=z+1\dfrac{x + 1}{7} = \dfrac{y + 1}{-6} = z + 1 and r⃗=(3i^+5j^+7k^)+μ(i^−2j^+k^)\vec{r} = (3\hat{i} + 5\hat{j} + 7\hat{k}) + \mu(\hat{i} - 2\hat{j} + \hat{k}) then [p][p] is... ,(where [ . ] denotes the greatest integer function.)

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  7. Question 7

    19 Apr 2026 S2

    93
    The shortest distance between the lines r⃗=(4i^−j^)+λ(i^+2j^−3k^)\vec{r} = (4\hat{i} - \hat{j}) + \lambda(\hat{i} + 2\hat{j} - 3\hat{k}) and r⃗=(i^−j^+2k^)+μ(i^+4j^−5k^)\vec{r} = (\hat{i} - \hat{j} + 2\hat{k}) + \mu(\hat{i} + 4\hat{j} - 5\hat{k}) is...

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  8. Question 8

    20 Apr 2026 S1

    94
    The shortest distance between the lines x−33=y−8−1=z−31\dfrac{x - 3}{3} = \dfrac{y - 8}{-1} = \dfrac{z - 3}{1} and x+3−3=y+72=z−64\dfrac{x + 3}{-3} = \dfrac{y + 7}{2} = \dfrac{z - 6}{4} is

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