𝑥−2 𝑦−4 𝑧−3 𝑥−2 𝑦−5 1−𝑧

13. Find the value of p, so that lines = = and = = are perpendicular to each
−2 3𝑝 4 4𝑝 2 7
other.
̂ + 𝜆(𝑖̂ + 2𝑗̂ − 3𝑘̂) and 𝑟⃗ = (𝑖̂ − 𝑗̂ + 2𝑘̂
14. Find the shortest distance between the lines 𝑟⃗ = (4𝑖̂ − 𝑗)
)+𝜇(2𝑖̂ + 4𝑗̂ − 5𝑘̂)
−𝑥+2 𝑦−1 𝑧+3 𝑥+2 2𝑦−8 𝑧−5
15. Find the angle between following pair of lines = = and = = and check
−2 7 −3 −1 4 4
whether the lines are parallel or perpendicular.
𝑥−11 𝑦+2 𝑧+8
16. Find the image of the point (2, -1, 5) in the line = = .
10 −4 −11
17. If vectors 𝑎⃗ = 2𝑖̂ + 2𝑗̂ + 3𝑘̂ , 𝑏⃗⃗ = −𝑖̂ + 2𝑗̂ + 𝑘̂ 𝑎𝑛𝑑 𝑐⃗ = 3𝑖̂ + 𝑗̂ are such that 𝑏⃗⃗ + 𝜆𝑐⃗ is perpendicular to
𝑎⃗ , then find the value of 𝜆.
18. The two co-initial adjacent sides of a parallelogram are 2𝑖̂ − 4𝑗̂ − 5𝑘̂ and 2𝑖̂ + 2𝑗̂ + 3𝑘̂ . Find its
diagonals and use them to find the area of the parallelogram.
19. An ant is moving along the vector 𝑙1 = 𝑖̂ − 2𝑗̂ + 3𝑘̂. Few sugar crystals are kept along the vector
𝑙2 = 3𝑖̂ − 2𝑗̂ + 𝑘̂ which are inclined at an angle 𝜃 with the vector 𝑙1 . Then find the angle 𝜃. Also, find
the scalar projection of 𝑙1 on 𝑙2 .
20. Find the vector and the cartesian equation of the line that passes through (-1, 2, 7) and is
perpendicular to the lines 𝑟⃗ = (2𝑖̂ + 𝑗̂ − 3𝑘) ̂ +𝜆(𝑖̂ + 2𝑗̂ + 5𝑘̂) and 𝑟⃗ = (3𝑖 ̂ + 3𝑗̂ − 7𝑘̂) + 𝜇(3𝑖̂ − 2𝑗̂ +
5𝑘̂).
21. Find the shortest distance between the lines 𝑟⃗ = (−𝑖̂ − 𝑗̂ − 𝑘̂) + 𝜆(7𝑖̂ − 6𝑗̂ + 𝑘̂) and 𝑟⃗ = (3𝑖̂ + 5𝑗̂ +
7𝑘̂ )+𝜇(𝑖̂ − 2𝑗̂ + 𝑘̂) , where 𝜆 and 𝜇 are parametres.
𝑥−3 𝑦+1 𝑧−1
22. Find the image of the point (1, 2, 1) with respect to the line 1 = 2 = 3 . Also find the equation
of the line joining the given point and its image .
23. Find a vector of magnitude 4 units perpendicular to each of the vectors 2𝑖̂ − 𝑗̂ + 𝑘̂ and 𝑖̂ + 𝑗̂ − 𝑘̂ and
hence verify your answer.
4−𝑥
24. Find the coordinates of the foot of the perpendicular drawn from the point (2, 3, -8) in the line 2 =
𝑦 1−𝑧
= . Also find the perpendicular distance of the given point from the line.
6 3
25. Find the shortest distance between the lines 𝑙1 and 𝑙2 given below
𝑥 𝑦 𝑧
𝑙1 : The line passing through (2, -1, 1) and parallel to 1 = 1 = 3 𝑙2 : 𝑟⃗ = 𝑖̂ + (2𝜇 + 1)𝑗̂ − (𝜇 + 2)𝑘̂.
26. Find the vector and cartesian equations of the line which is perpendicular to the lines with
𝑥+2 𝑦−3 𝑧+1 𝑥−1 𝑦−2 𝑧−3
equations 1 = 2 = 4 and 2 = 3 = 4 and passes through the point (1, 1, 1) . Also, find
the angle between the given lines.
27. Find the shortest distance between the lines given by 𝑟⃗ = (2 + 𝜆)𝑖̂ − (3 + 𝜆)𝑗̂ + (5 + 𝜆)𝑘̂) and 𝑟⃗ =
(2𝜇 − 1)𝑖̂ + (4𝜇 − 1)𝑗̂ + (5 − 3𝜇)𝑘̂ ).
𝑥+2 𝑦−1 𝑧−0
28. Vertices B and C of ∆𝐴𝐵𝐶 lie along the line = = . Find the area of the triangle given that
2 1 4
A has coordinates (1, -1, 2) and line segment BC has length 5 units.
⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
29. For any two vectors 𝑎⃗ and 𝑏⃗⃗, show that (1 + |𝑎⃗|2 ) (1 + |𝑏⃗⃗|2 ) =(1 − 𝑎⃗. 𝑏⃗⃗)2 +|𝑎⃗ + 𝑏⃗⃗ + (𝑎 × 𝑏⃗⃗)|2
30. If 𝑎⃗ = 𝑖̂ + 𝑗̂ + 𝑘̂ and 𝑏⃗⃗ = 𝑗̂ − 𝑘̂, then find a vector 𝑐⃗ such that 𝑎⃗ × 𝑐̂ = 𝑏̂ and 𝑎⃗. 𝑐⃗ = 3.
1−𝑥 7𝑦−14 𝑧−3 7−7𝑥 𝑦−5 6−𝑧
31. Find the value of p, so that the lines 𝑙1 : 3 = 𝑝 = 2 and 𝑙2 : 3𝑝 = 1 = 5 are
perpendicular to each other. Also, find the equation of a line passing through a point (3, 2, -4) and
parallel to the line 𝑙1 .
32. Find the value of 𝜆, so that the vectors 𝑎⃗ = 3𝑖̂ + 2𝑗̂ + 9𝑘̂ and 𝑏⃗⃗ = 𝑖̂ + 𝜆𝑗̂ + 3𝑘̂ are perpendicular to
each other.
33. Find the area of the triangle whose two sides are represented by the vectors 2𝑖̂ and -3𝑗̂.
𝑥−4 𝑦+1 𝑧 𝑥−1 𝑦−2 𝑧−3
34. Find the shortest distance between the lines 4 = 2 = 1 and 5 = 2 = 1 .
35. Find the equation of the line passing through the point (1, 2, -4) and perpendicular to the two lines
𝑟⃗ = (8𝑖̂ − 19𝑗̂ + 10𝑘) ̂ +𝜆(3𝑖̂ − 16𝑗̂ + 7𝑘̂) and 𝑟⃗ = (15𝑖̂ + 29𝑗̂ + 5𝑘̂) + 𝜇(3𝑖̂ + 8𝑗̂ − 5𝑘̂).
36. Find a vector of magnitude 5 units, perpendicular to each of the vectors (𝑎⃗+ 𝑏⃗⃗) and (𝑎⃗- 𝑏⃗⃗), where
𝑎⃗ = 𝑖̂ + 𝑗̂ + 𝑘̂ and 𝑏⃗⃗ = 𝑖̂ + 2𝑗̂ + 3𝑘̂ .
37. If 𝑎⃗ = 𝑖̂ + 2𝑗̂ + 𝑘̂ and 𝑏⃗⃗ = 2𝑖̂ + 𝑗̂ and 𝑐⃗ = 3𝑖̂ − 4𝑗̂ − 5𝑘̂ , then find a unit vector perpendicular to both
of the vectors (𝑎⃗- 𝑏⃗⃗) and (𝑐⃗- 𝑏⃗⃗) .
38. Find the position vector of a point C which divides the line segment joining A and B whose position
vectors are 2𝑎⃗+ 𝑏⃗⃗ and 𝑎⃗- 3𝑏⃗⃗, externally in the ratio 1:2. Also, show that A is the mid-point of the line
segment BC.
39. Show that the area of a parallelogram having diagonals 3𝑖̂ + 𝑗̂ − 2𝑘̂ and 𝑖̂ − 3𝑗̂ + 4𝑘̂ is 5√3 sq units.
40. If A(3, 5, -4), B(-1, 1, 2) andC(-5, -5, -2) are the vertices of a ∆𝐴𝐵𝐶 , then find the direction cosines
of AB, AC and BC.