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Feet of the common perpendicular (line of shortest distance) between two skew lines

This type of question, asked 4 times: 29 Jan 2024, 1 Feb 2024, 5 Apr 2024, 22 Jan 2025. All 4 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    29 Jan 2024

    176
    Let OO be the origin, and MM and NN be the points on the lines x−54=y−41=z−53\frac{x- 5}{4}=\frac{y- 4}{1}=\frac{z- 5}{3} and x+812=y+25=z+119\frac{x + 8}{12}=\frac{y + 2}{5}=\frac{z + 11}{9} respectively such that MNMN is the shortest distance between the given lines. Then OM→⋅ON→\overrightarrow{OM}\cdot\overrightarrow{ON} is equal to

  2. Question 2

    1 Feb 2024

    89
    Let the line of the shortest distance between the lines L1:r→=(i^+2j^+3k^)+λ(i^−j^+k^)L_{1}:\overrightarrow{r}= (\widehat{i}+ 2\widehat{j}+ 3\widehat{k}) + \lambda(\widehat{i}-\widehat{j}+\widehat{k}) and
    L2:r→=(4i^+5j^+6k^)+μ(i^+j^−k^)L_{2}:\overrightarrow{r}= (4\widehat{i}+ 5\widehat{j}+ 6\widehat{k}) + \mu(\widehat{i}+\widehat{j}-\widehat{k})
    intersect L1L_{1} and L2L_{2} at PP and QQ respectively. If (α,β,γ)(\alpha,\beta,\gamma) is the midpoint of the line segment PQPQ, then 2(α+β+γ)2(\alpha + \beta + \gamma) is equal to

  3. Question 3

    5 Apr 2024

    174
    Let the point (−1,α,β)( - 1,\alpha,\beta) lie on the line of the shortest distance between the lines x+2−3=y−24=z−52\frac{x+ 2}{- 3}=\frac{y- 2}{4}=\frac{z- 5}{2} and  x+2−1=y+62=z−10\ \frac{x+ 2}{- 1}=\frac{y+ 6}{2}=\frac{z- 1}{0}. Then (α−β)2(\alpha-\beta)^{2} is equal to.

  4. Question 4

    22 Jan 2025

    58
    Let L1:x−12=y−23=z−34L_{1}:\frac{x- 1}{2}=\frac{y- 2}{3}=\frac{z- 3}{4} and L2:x−23=y−44=z−55L_{2}:\frac{x- 2}{3}=\frac{y- 4}{4}=\frac{z- 5}{5} be two lines. Then which of the following points lies on the line of the shortest distance between L1L_{1} and L2L_{2} ?

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