More practice questions

5. Q5. For the lines of NCERT Example 9 (page 388), r = î + ĵ + λ(2î − ĵ + k̂) and r = 2î + ĵ − k̂ + μ(3î − 5ĵ + 2k̂), find the feet of the common perpendicular.

   Show answerHide answer

   The conditions are 6λ − 13μ = 1 and 13λ − 38μ = 1, so λ = 25/59 and μ = 7/59. P = (109/59, 34/59, 25/59) and Q = (139/59, 24/59, −45/59). PQ = (10/59)(3î − ĵ − 7k̂), parallel to b_₁ × b_₂, with length 10/√59.

6. Q6. For the lines of NCERT Exercise 11.2, Q15 (page 390), find the feet of the common perpendicular and check the distance 8/√29.

   Show answerHide answer

   With a_₁ = î − 2ĵ + 3k̂, b_₁ = −î + ĵ − 2k̂, a_₂ = î − ĵ − k̂ and b_₂ = î + 2ĵ − 2k̂, the conditions give λ = 31/29 and μ = −15/29 (λ = t and μ = s in NCERT’s letters). P = (−2/29, −27/29, 25/29) and Q = (14/29, −59/29, 1/29), so PQ = (8/29)(2î − 4ĵ − 3k̂), of length 8/√29.

7. Q7. Show that the lines (x − 1)/1 = y/1 = z/0 and x/0 = y/1 = (z − 3)/1 are skew, and find their common perpendicular.

   Show answerHide answer

   The directions 1, 1, 0 and 0, 1, 1 are not proportional, so the lines are not parallel. P = (1 + λ, λ, 0) and Q = (0, μ, 3 + μ), so PQ = (−1 − λ)î + (μ − λ)ĵ + (3 + μ)k̂. The conditions −1 − 2λ + μ = 0 and 2μ − λ + 3 = 0 give λ = −5/3 and μ = −7/3. P = (−2/3, −5/3, 0) and Q = (0, −7/3, 2/3), so PQ = (2/3)(î − ĵ + k̂) ≠ 0: the lines do not meet, so they are skew, d = 2/√3, and the common perpendicular is (x + 2/3)/1 = (y + 5/3)/(−1) = z/1.

8. Q8. What happens to the two conditions when the lines are parallel?

   Show answerHide answer

   With b_₂ = b_₁, both conditions become the same equation in λ − μ, so there are infinitely many common perpendiculars, all of the same length: the distance between the parallel lines.