JEE Main - Shortest distance between skew lines Question Bank with Options

1. The shortest distance between the lines x = 1, y = 2, z = 3 and x = 1, y = 2, z = 3 is:

A) 0

B) 1

C) 2

D) 3

2. If the shortest distance between two skew lines is given by d = |(a2 - a1) . (b1 x b2)| / |b1 x b2|, and (a2 - a1) is perpendicular to (b1 x b2), what is the shortest distance?

A) 0

B) 1

C) Depends on |b1 x b2|

D) Depends on |a2 - a1|

3. What is the magnitude of the vector cross product of two non-parallel vectors?

A) It is zero if the vectors are parallel

B) It is always positive

C) It is equal to the dot product

D) It is zero if the vectors are perpendicular

4. If the lines are given by r = a1 + λb1 and r = a2 + μb2, and b1 is parallel to b2, then the lines are:

A) Skew

B) Intersecting

C) Parallel

D) Coincident

5. The shortest distance between the lines x = 3, y = 2, z = 1 + t and x = 1, y = 1, z = 2 + s is:

A) √2

B) √3

C) √5

D) √6

6. What does it mean if the shortest distance between two lines is calculated using the formula for parallel lines, but the direction vectors are not parallel?

A) The lines are skew

B) The lines intersect

C) The lines are coincident

D) The calculation is invalid

7. The shortest distance between the lines r = (2i - j + k) + λ(i - 2j + k) and r = (i - j + 2k) + μ(2i + j - k) is:

A) 3/√14

B) 4/√14

C) 5/√14

D) 6/√14

8. If the lines r = a1 + λb1 and r = a2 + μb2 are perpendicular, the shortest distance is:

A) |(a2 - a1) . (b1 x b2)| / |b1 x b2|

B) 0

C) |(a2 - a1) . b1| / |b1|

D) |(a2 - a1) . b2| / |b2|

9. The shortest distance between the lines x = 1 + 2t, y = 2 + 3t, z = 3 + 4t and x = 2 + 3s, y = 4 + 4s, z = 5 + 5s is:

A) 0

B) 1

C) 2

D) 3

10. Which of the following pairs of lines are skew?

A) r = i + λj and r = 2i + μj

B) r = i + λj and r = i + μ(2j)

C) r = i + λj and r = 2i + μ(i+j)

D) r = i + λj and r = i + λ(2j)

11. The shortest distance between the origin and the line r = (i + j + k) + λ(2i + 3j + 4k) is:

A) √14 / √29

B) √29 / √14

C) √3 / √29

D) √29 / 3

12. If two lines are parallel, with direction vector b, and points a1 and a2, the shortest distance is the length of the projection of (a2 - a1) onto:

A) b

B) a1 x a2

C) a2 - a1

D) b x (a2 - a1)

13. What is the shortest distance between the lines x = 1, y = 1, z = 1+t and x = 2, y = 2, z = 2+s?

A) √2

B) √3

C) √5

D) √6

14. If the shortest distance between two skew lines is d, and the angle between the vector (a2 - a1) and (b1 x b2) is α, then:

A) d = |(a2 - a1) . (b1 x b2)| / |b1 x b2|

B) d = |a2 - a1| cos(α)

C) d = |b1 x b2| sin(α)

D) d = |a2 - a1| / cos(α)

15. What is the shortest distance between the lines x = t, y = 0, z = 0 and x = 0, y = s, z = 0?

A) 0

B) 1

C) 2

D) 3

16. Consider the lines L1: r = i + 2j + k + λ(2i - j + k) and L2: r = 2i + j - k + μ(i + j - k). What is the shortest distance between the points on these lines that are closest to each other?

A) 4/√22

B) 3/√22

C) 5/√22

D) 6/√22

17. The shortest distance between two skew lines is always:

A) Positive

B) Non-negative

C) Zero

D) Negative

18. Which of the following is NOT a method to find the shortest distance between skew lines?

A) Vector method using cross product

B) Cartesian method using determinant

C) Finding the intersection point and measuring distance

D) Projection of a connecting vector onto the common perpendicular

19. If b1 and b2 are the direction vectors of two skew lines, the direction vector of the common perpendicular is proportional to:

A) b1 x b2

B) b1 + b2

C) b1 - b2

D) b1

20. The shortest distance between the lines x = 1 + t, y = 2 + t, z = 3 + t and x = 2 + s, y = 3 + s, z = 4 + s is:

A) 0

B) 1

C) 2

D) 3

21. Find the shortest distance between the lines (x-1)/2 = (y-2)/3 = (z-3)/4 and (x-2)/3 = (y-4)/4 = (z-5)/5.

A) 1/√2

B) 1/√3

C) 1/√5

D) 1/√6

22. What is the condition for the shortest distance between two lines r = a1 + λb1 and r = a2 + μb2 to be zero?

A) b1 x b2 = 0

B) (a2 - a1) . (b1 x b2) = 0

C) (a2 - a1) . (b1 x b2) != 0

D) b1 and b2 are parallel

23. If two lines are given by x = 2y = 3z and x+1 = y-2 = z+3, what is the shortest distance between them?

A) 0

B) 1/√2

C) √2

D) 2

24. Let the equations of two skew lines be r = a1 + λb1 and r = a2 + μb2. The shortest distance is the length of the projection of the vector (a2 - a1) onto which vector?

A) b1 x b2

B) b1 + b2

C) b1 - b2

D) b1

25. The shortest distance between the lines r = (3i + 2j - 4k) + λ(i + 2j + 2k) and r = (5i - 2j + k) + μ(3i + 2j + k) is:

A) √58 / √29

B) √29 / √58

C) 3/√29

D) √29/3

26. If the shortest distance between two skew lines is given by d, and the angle between their direction vectors is θ, what is the relationship between d and cos(θ)?

A) No direct relationship

B) d is proportional to cos(θ)

C) d is inversely proportional to cos(θ)

D) d is proportional to sin(θ)

27. What is the shortest distance between the y-axis and the line x=1, y=t, z=2?

A) √5

B) √3

C) √2

D) √6

28. The vector form of a line is r = a + λb. If a line passes through the origin, what is the value of vector a?

A) 0 (zero vector)

B) i

C) j

D) k

29. If a1, b1, c1 and a2, b2, c2 are the direction ratios of two lines, and x1, y1, z1 and x2, y2, z2 are points on them, the shortest distance is zero if:

A) The lines are intersecting

B) The lines are parallel

C) The lines are coincident

D) The lines are perpendicular

30. What is the shortest distance between the lines x = 1, y = 2, z = 3 + t and x = 4, y = 5, z = 6 + s?

A) √27

B) √30

C) √22

D) √18

31. Consider the lines x = 1, y = 2, z = 3 + t and x = 4, y = 5, z = 6 + s. These lines are:

A) Skew

B) Parallel

C) Intersecting

D) Coincident

32. What is the formula for the shortest distance between two parallel lines r = a1 + λb and r = a2 + μb?

A) |(a2 - a1) x b| / |b|

B) |(a1 - a2) x b| / |b|

C) |(a2 - a1) . b| / |b|

D) |(a1 - a2) . b| / |b|

33. Two lines are given by r = (i + j) + λ(2i - j + k) and r = (2i + j - k) + μ(i + j - k). Are these lines skew?

A) Yes, because their direction vectors are not parallel

B) No, because their direction vectors are not parallel

C) Yes, because the vector connecting their points is not parallel to their direction vectors

D) No, because the vector connecting their points is parallel to their direction vectors

34. If the direction ratios of two lines are proportional, what is the shortest distance between them if they are not coincident?

A) The distance between the points where a transversal intersects them

B) Infinite

C) Zero

D) Depends on the intercept

35. The shortest distance between the lines x/1 = y/2 = z/3 and x = 3t, y = 4t, z = 5t is:

A) 0

B) 1

C) 2

D) 3

36. What is the determinant form to calculate the shortest distance between two skew lines given by their symmetric equations?

A) | x2-x1 y2-y1 z2-z1 | / sqrt( (b1c2-b2c1)^2 + (c1a2-c2a1)^2 + (a1b2-a2b1)^2 )

B) | a1 b1 c1 |

C) | a2 b2 c2 |

D) | x2-x1 y2-y1 z2-z1 |

37. If two lines are given in Cartesian form as (x-x1)/a1 = (y-y1)/b1 = (z-z1)/c1 and (x-x2)/a2 = (y-y2)/b2 = (z-z2)/c2, what is the formula for the shortest distance?

A) |( (x2-x1, y2-y1, z2-z1) . (a1, b1, c1) x (a2, b2, c2) )| / |(a1, b1, c1) x (a2, b2, c2)|

B) |( (x1-x2, y1-y2, z1-z2) . (a1, b1, c1) x (a2, b2, c2) )| / |(a1, b1, c1) x (a2, b2, c2)|

C) |( (x2-x1, y2-y1, z2-z1) x (a1, b1, c1) . (a2, b2, c2) )| / |(a1, b1, c1) x (a2, b2, c2)|

D) |( (x2-x1, y2-y1, z2-z1) . (a1, b1, c1) + (a2, b2, c2) )| / |(a1, b1, c1) x (a2, b2, c2)|

38. What is the shortest distance between the lines x=1+2t, y=2-t, z=t and x=2+s, y=1+s, z=-s?

A) 3/√14

B) 4/√14

C) 5/√14

D) 6/√14

39. Calculate the shortest distance between the lines L1: r = (i + 2j + k) + λ(2i - j + k) and L2: r = (2i + j - k) + μ(i + j - k).

A) 3/√22

B) 4/√22

C) 5/√22

D) 6/√22

40. Calculate the magnitude of (b1 x b2) for the lines L1: r = (i + 2j + k) + λ(2i - j + k) and L2: r = (2i + j - k) + μ(i + j - k).

A) √22

B) √19

C) √30

D) √27

41. For the lines L1: r = (i + 2j + k) + λ(2i - j + k) and L2: r = (2i + j - k) + μ(i + j - k), what is the vector (b1 x b2)?

A) 2i + 3j + 3k

B) 2i - 3j + 3k

C) 2i + 3j - 3k

D) -2i + 3j + 3k

42. Consider the lines L1: r = (i + 2j + k) + λ(2i - j + k) and L2: r = (2i + j - k) + μ(i + j - k). What is the vector (a2 - a1)?

A) i - j - 2k

B) i + j + 2k

C) -i + j + 2k

D) i + j - 2k

43. If the shortest distance between two skew lines is zero, what does it imply?

A) The lines are parallel

B) The lines intersect

C) The lines are coincident

D) The lines are perpendicular

44. What is the condition for two lines r = a1 + λb1 and r = a2 + μb2 to be skew?

A) b1 and b2 are parallel and a2 - a1 is not parallel to b1

B) b1 and b2 are not parallel and a2 - a1 is not parallel to b1 or b2

C) b1 and b2 are not parallel and (a2 - a1) . (b1 x b2) = 0

D) b1 and b2 are parallel and (a2 - a1) . (b1 x b2) != 0

45. In the shortest distance formula for skew lines, what does b1 x b2 represent?

A) A vector parallel to both b1 and b2

B) A vector perpendicular to both b1 and b2

C) The sum of the direction vectors

D) The difference of the direction vectors

46. For skew lines r = a1 + λb1 and r = a2 + μb2, what does the term (a2 - a1) represent in the shortest distance formula?

A) The vector connecting a point on the first line to a point on the second line

B) The direction vector of the first line

C) The direction vector of the second line

D) The vector perpendicular to both direction vectors

47. If two lines are given by vector equations r = a1 + λb1 and r = a2 + μb2, what is the formula for the shortest distance between them?

A) |(a2 - a1) . (b1 x b2)| / |b1 x b2|

B) |(a1 - a2) . (b1 x b2)| / |b1 x b2|

C) |(a2 - a1) x (b1 . b2)| / |b1 x b2|

D) |(a1 - a2) x (b1 x b2)| / |b1 x b2|

48. What is the geometric interpretation of the shortest distance between two skew lines?

A) The length of the common perpendicular segment

B) The distance from one line to a plane containing the other

C) The minimum distance between any two points on the lines

D) The distance measured along the x-axis

49. What type of lines are necessary to calculate the shortest distance between them?

A) Parallel lines

B) Intersecting lines

C) Skew lines

D) Concurrent lines

Shortest distance between skew lines - Question Bank