Three Dimensional Geometry Class 12 Notes

• Angle between two lines follows from their direction ratios via the dot-product formula.
• Angle between two planes is the angle between their normals.
• Angle between a line and a plane is the complement of the angle between the line and the plane's normal.
• Two lines are skew when they are neither parallel nor intersecting; their shortest distance uses a common perpendicular.

Example: Find the direction cosines of the line with direction ratios 3, 4, 12.

Solution: Magnitude $=\sqrt{9+16+144}=13$ (note 3-4-12-13), so the direction cosines are 3/13, 4/13, 12/13, and $(3/13{)}^2+(4/13{)}^2+(12/13{)}^2=1$.

Example: Find the perpendicular distance of $(1,1,1)$ from the plane $2x-y+2z=6$.

Solution: $d=\frac{|2(1)-1+2(1)-6|}{\sqrt{4+1+4}}=\frac{|-3|}{3}=1$ unit.

Exercises (NCERT Q1–Q59)

• 1) If a line makes angles with the x, y and z-axes respectively, find its direction cosines.
• 2) Find the direction cosines of a line which makes equal angles with the coordinate axes.
• 3) A line makes an angle of 90° and 60° respectively with the positive directions of the x and y-axes. Find the angle it makes with the positive direction of the z-axis.
• 4) Show that the points (2,3,4), (−1,−2,1) and (5,8,7) are collinear.
• 5) Find the direction cosines of the side AB of the triangle whose vertices are A(3,5,−4), B(−1,1,2) and C(−5,−5,−2).
• 6) Find the direction cosines of the side BC of the triangle whose vertices are A(3,5,−4), B(−1,1,2) and C(−5,−5,−2).
• 7) Find the direction cosines of the side CA of the triangle whose vertices are A(3,5,−4), B(−1,1,2) and C(−5,−5,−2).
• 8) Show that the three lines with direction cosines proportional to (12,−3,−4), (4,12,3) and (3,−4,12) are mutually perpendicular.
• 9) Show that the line through the points (1,−1,2) and (3,4,−2) is perpendicular to the line through the points (0,3,2) and (3,5,6).
• 10) Show that the line through the points (4,7,8) and (2,3,4) is parallel to the line through the points (−1,−2,1) and (1,2,5).
• 11) Find the equation of the line which passes through the point (1,2,3) and is parallel to the vector . Write its vector equation.
• 12) Find the cartesian equation of the line which passes through the point (1,2,3) and is parallel to the vector .
• 13) Find the equation of the line, in vector and cartesian form, that passes through the point with position vector and is in the direction .
• 14) Find the cartesian equation of the line that passes through the point (−2,4,−5) and is parallel to the line .
• 15) The cartesian equation of a line is . Write its vector form.
• 16) Find the vector equation of the line passing through the points (2,−1,−3) and (−2,2,5).
• 17) Find the cartesian equation of the line passing through the points (2,−1,−3) and (−2,2,5).
• 18) Find the vector and cartesian equations of the line passing through the points A(3,4,−6) and B(1,2,−7).
• 19) Find the angle between the pair of lines and .
• 20) Find the angle between the pair of lines and .
• 21) Find the values of p so that the lines and are at right angles.
• 22) Show that the lines and are perpendicular to each other.
• 23) Find the shortest distance between the lines and .
• 24) Find the shortest distance between the lines and .
• 25) Find the shortest distance between the lines whose vector equations are and .
• 26) Find the shortest distance between the lines whose vector equations are and .
• 27) Determine the direction cosines of the normal to the plane and the distance from the origin: .
• 28) Determine the direction cosines of the normal to the plane and the distance from the origin: .
• 29) Determine the direction cosines of the normal to the plane and the distance from the origin: .
• 30) Determine the direction cosines of the normal to the plane and the distance from the origin: .
• 31) Find the vector equation of a plane which is at a distance of 7 units from the origin and normal to the vector .
• 32) Find the cartesian equation of the plane .
• 33) Find the cartesian equation of the plane .
• 34) Find the cartesian equation of the plane .
• 35) Find the coordinates of the foot of the perpendicular drawn from the origin to the plane .
• 36) Find the coordinates of the foot of the perpendicular drawn from the origin to the plane .
• 37) Find the coordinates of the foot of the perpendicular drawn from the origin to the plane .
• 38) Find the coordinates of the foot of the perpendicular drawn from the origin to the plane .
• 39) Find the vector and cartesian equations of a plane which passes through the point (1,0,−2) and has normal vector .
• 40) Find the equation of the plane through the points (1,1,−1), (6,4,−5) and (−4,−2,3).
• 41) Find the equation of the plane through the points (1,1,0), (1,2,1) and (−2,2,−1).
• 42) Find the intercepts cut off by the plane .
• 43) Find the equation of the plane with intercept 3 on the y-axis and parallel to the ZOX plane.
• 44) Find the equation of the plane through the intersection of the planes and and the point (2,2,1).
• 45) Find the vector equation of the plane passing through the intersection of the planes and and through the point (2,1,3).
• 46) Find the angle between the planes and .
• 47) Find the angle between the planes and .
• 48) Find the angle between the planes and .
• 49) Find the angle between the planes whose vector equations are and .
• 50) Determine whether the planes and are parallel or perpendicular, and find the angle between them.
• 51) Determine whether the planes and are parallel or perpendicular, and find the angle between them.
• 52) Determine whether the planes and are parallel or perpendicular, and find the angle between them.
• 53) Determine whether the planes and are parallel or perpendicular, and find the angle between them.
• 54) Find the distance of the point (0,0,0) from the plane .
• 55) Find the distance of the point (3,−2,1) from the plane .
• 56) Find the distance of the point (2,3,−5) from the plane .
• 57) Find the distance of the point (−6,0,0) from the plane .
• 58) Find the distance of the point (−1,−5,−10) from the point of intersection of the line and the plane .
• 59) Find the equation of the plane through the intersection of the planes and , which is perpendicular to the plane .

Answers at a glance

1. 1
   Q1 → Direction cosines are (0, −1/√2, 1/√2).
2. 2
   Q2 → (±1/√3, ±1/√3, ±1/√3); taking the acute direction, (1/√3, 1/√3, 1/√3).
3. 3
   Q3 → The angle with the positive z-axis is 30° or 150°.
4. 4
   Q4 → Yes, the three points are collinear.
5. 5
   Q5 → AB: (−2/√17, −2/√17, 3/√17).
6. 6
   Q6 → BC: (−2/√17, −3/√17, −2/√17).
7. 7
   Q7 → CA: (4/√42, 5/√42, −1/√42).
8. 8
   Q8 → Each pair is perpendicular, so the three lines are mutually perpendicular.
9. 9
   Q9 → The two lines are perpendicular.
10. 10
    Q10 → The two lines are parallel.
11. 11
    Q11 → answer not in source — verify against the textbook
12. 12
    Q12 → answer not in source — verify against the textbook
13. 13
    Q13 → Vector: r = (2i − j + 4k) + λ(i + 2j − k). Cartesian: (x−2)/1 = (y+1)/2 = (z−4)/(−1).
14. 14
    Q14 → answer not in source — verify against the textbook
15. 15
    Q15 → answer not in source — verify against the textbook
16. 16
    Q16 → answer not in source — verify against the textbook
17. 17
    Q17 → answer not in source — verify against the textbook
18. 18
    Q18 → Vector: r = (3i + 4j − 6k) + λ(−2i − 2j − k). Cartesian: (x−3)/(−2) = (y−4)/(−2) = (z+6)/(−1).
19. 19
    Q19 → θ = cos⁻¹(19/21).
20. 20
    Q20 → θ = cos⁻¹(19/21).
21. 21
    Q21 → p = 70/11.
22. 22
    Q22 → The lines are perpendicular to each other.
23. 23
    Q23 → Shortest distance = 3/√2.
24. 24
    Q24 → Shortest distance = 2√29.
25. 25
    Q25 → Shortest distance = 3/√19.
26. 26
    Q26 → Shortest distance = 8/√29.
27. 27
    Q27 → Direction cosines (0,0,1); distance 2.
28. 28
    Q28 → Direction cosines (1/√3, 1/√3, 1/√3); distance 1/√3.
29. 29
    Q29 → Direction cosines (2/√14, 3/√14, −1/√14); distance 5/√14.
30. 30
    Q30 → Direction cosines (0,1,0); distance 8/5.
31. 31
    Q31 → answer not in source — verify against the textbook
32. 32
    Q32 → x + y − z = 2.
33. 33
    Q33 → 2x + 3y − 4z = 1.
34. 34
    Q34 → (s − 2t)x + (3 − t)y + (2s + t)z = 15.
35. 35
    Q35 → (24/29, 36/29, 48/29).
36. 36
    Q36 → (0, 18/25, 24/25).
37. 37
    Q37 → (1/3, 1/3, 1/3).
38. 38
    Q38 → (0, −8/5, 0).
39. 39
    Q39 → r·(i + j − k) = 3, i.e. x + y − z = 3.
40. 40
    Q40 → The points are collinear, so infinitely many planes pass through them.
41. 41
    Q41 → 2x + 3y − 3z = 5.
42. 42
    Q42 → Intercepts are 5/2, 5, −5 on the x, y, z axes respectively.
43. 43
    Q43 → y = 3.
44. 44
    Q44 → 7x − 5y + 4z − 8 = 0.
45. 45
    Q45 → answer not in source — verify against the textbook
46. 46
    Q46 → θ = cos⁻¹(15/√731).
47. 47
    Q47 → θ = cos⁻¹(5/√58).
48. 48
    Q48 → θ = cos⁻¹(11/(7√14)).
49. 49
    Q49 → θ = 90°, i.e. the planes are perpendicular.
50. 50
    Q50 → Neither parallel nor perpendicular; angle θ = cos⁻¹(2/5).
51. 51
    Q51 → Perpendicular.
52. 52
    Q52 → Parallel; angle 0°.
53. 53
    Q53 → Parallel; angle 0°.
54. 54
    Q54 → 3/13.
55. 55
    Q55 → 13/3.
56. 56
    Q56 → 3.
57. 57
    Q57 → 2.
58. 58
    Q58 → 13 units.
59. 59
    Q59 → 51x + 15y − 50z + 173 = 0.

MCQ Questions

1 Mark Questions

1. 1
   Write the direction cosines of the x-axis. (Ans: (1, 0, 0))
2. 2
   Find the distance of the origin from the plane ax + by + cz = d. (Ans: |d|/√(a² + b² + c²))
3. 3
   State the condition for two lines in space to be coplanar.
4. 4
   What is the cartesian equation of a line through (x₁,y₁,z₁) with direction ratios (a, b, c)?

2 Mark Questions

1. 1
   Find the direction cosines of the vector joining (1, 0) and (2, 3) taken in 2D as a direction-ratio example. (Ans: if in 2D components (1,3), l = 1/√10, m = 3/√10)
2. 2
   Show that the plane x − y + z = 4 and the line with direction ratios (0, 1, 1) are not perpendicular.
3. 3
   Find the angle between the planes x + y + z = 1 and 2x − y + 3z = 5.

3 Mark Questions

1. 1
   Find the shortest distance between the skew lines r = (i + 2j − k) + λ(i − j + k) and r = 2i − j − k + μ(2i + j + 2k), using the formula with the scalar triple product. (Ans: distance = |(a₂ − a₁)·(b₁ × b₂)|/|b₁ × b₂| = 3/√59)
2. 2
   Find the equation of the plane through the point (1, 2, 3) and perpendicular to the line with direction ratios (2, −1, 3), in both vector and cartesian form.
3. 3
   Find the image of the point (1, 2, 3) in the plane x + y + z = 6.

4 Mark Questions

1. 1
   Find the perpendicular distance of the point (1, 2, 3) from the plane x + 2y − 2z = 9, then find the foot of the perpendicular and the image of the point. Give all three answers.
2. 2
   Show that the lines (x − 1)/2 = (y − 2)/3 = (z − 3)/4 and (x − 2)/4 = (y − 3)/6 = (z − 5)/8 are coplanar and find the equation of the plane containing them. Hence (or otherwise) find the shortest distance for parallel lines.