Q5. Beyond the NCERT text (JEE): find k so that (x − 1)/2 = (y + 1)/3 = (z − 1)/4 and (x − 3)/1 = (y − k)/2 = z/1 intersect.
b × b = −5î + 2ĵ + k̂ ≠ 0, and a − a = 2î + (k + 1)ĵ − k̂. The lines meet when the dot product −10 + 2(k + 1) − 1 = 2k − 9 is 0, that is, k = 9/2.
Q6. Find the distance between the parallel lines (x − 1)/2 = (y − 2)/3 = (z − 3)/6 and x/2 = (y + 1)/3 = (z − 1)/6.
b = 2î + 3ĵ + 6k̂, |b| = 7, and a − a = −î − 3ĵ − 2k̂. Then b × (a − a) = 12î − 2ĵ − 3k̂, of length √157, so d = √157/7.
Q7. Show that (x − 5)/4 = (y − 7)/4 = (z + 3)/(−5) and (x − 8)/7 = (y − 4)/1 = (z − 5)/3 intersect, and find the point.
b × b = 17î − 47ĵ − 24k̂ and a − a = 3î − 3ĵ + 8k̂; the dot product is 51 + 141 − 192 = 0. From (5 + 4λ, 7 + 4λ, −3 − 5λ) = (8 + 7μ, 4 + μ, 5 + 3μ): 4λ − 7μ = 3 and 4λ − μ = −3 give μ = −1 and λ = −1. The point is (1, 3, 2), and the z-coordinates agree.
Q8. Find the shortest distance between the z-axis and the line through (1, 0, 0) parallel to ĵ + k̂.
b × b = k̂ × (ĵ + k̂) = −î, of length 1, and a − a = î, so d = |−1| = 1.
Q9. For lines r = a + λb and r = a + μb with b × b = 0 and a − a not parallel to b, what can you say?
They are parallel and distinct, so they never meet, but they are not skew: parallel lines lie in one plane. Use the parallel-lines formula for the distance.
Q10. Can two skew lines be 0 apart? Can two parallel lines be?
Skew lines: no, since a distance of 0 would mean a common point. Parallel lines: only if they coincide, when b × (a − a) = 0.
Q11. Find the shortest distance between the lines r = 6î + 2ĵ + 2k̂ + λ(î − 2ĵ + 2k̂) and r = −4î − k̂ + μ(3î − 2ĵ − 2k̂) (NCERT Miscellaneous Exercise on Chapter 11, Q4, page 391).
b × b = 8î + 8ĵ + 4k̂, of length 12, and a − a = −10î − 2ĵ − 3k̂. The dot product is −80 − 16 − 12 = −108, so d = 108/12 = 9.