Class 12 Mathematics Notes Chapter 5 (Three Dimensional Geometry) – Mathematics Part-II Book

Detailed Notes with MCQs of Chapter 11: Three Dimensional Geometry from your NCERT Part-II book. This is a very important chapter, not just for your board exams but also extensively featured in various government entrance examinations. It builds upon your knowledge of vectors and introduces the geometry of lines and planes in 3D space. Pay close attention to the concepts and formulae.

Chapter 11: Three Dimensional Geometry - Detailed Notes

1. Introduction & Basic Concepts

• Coordinate Axes & Planes: In 3D space, we use three mutually perpendicular lines (x-axis, y-axis, z-axis) intersecting at the origin O(0,0,0). These axes define three coordinate planes (XY-plane, YZ-plane, ZX-plane).
• Coordinates of a Point: Any point P in space is represented by an ordered triplet (x, y, z).
• Distance Formula: The distance between two points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) is given by:
  PQ = √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²]
• Section Formula:
  • Internal Division: The coordinates of point R that divides the line segment joining P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) internally in the ratio m:n are:
    R = ( (mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n), (mz₂ + nz₁)/(m+n) )
  • External Division: The coordinates of point R that divides the line segment joining P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) externally in the ratio m:n are:
    R = ( (mx₂ - nx₁)/(m-n), (my₂ - ny₁)/(m-n), (mz₂ - nz₁)/(m-n) )
  • Mid-point: If R is the midpoint (ratio 1:1), its coordinates are:
    R = ( (x₁ + x₂)/2, (y₁ + y₂)/2, (z₁ + z₂)/2 )

2. Direction Cosines (DCs) and Direction Ratios (DRs) of a Line

• Direction Angles: The angles α, β, γ made by a directed line L with the positive directions of the x, y, and z axes respectively are called direction angles. (0 ≤ α, β, γ ≤ π)
• Direction Cosines (DCs): If α, β, γ are the direction angles, then their cosines, cos α, cos β, cos γ, are the Direction Cosines of the line L. They are usually denoted by l, m, n.
  • l = cos α, m = cos β, n = cos γ
• Key Relation: For any line with DCs l, m, n:
  l² + m² + n² = 1 (This is fundamental!)
• Direction Ratios (DRs): Any three numbers a, b, c proportional to the direction cosines l, m, n are called Direction Ratios.
  • l/a = m/b = n/c = k (for some non-zero constant k)
  • a = kl, b = km, c = kn
• Finding DCs from DRs: If a, b, c are DRs of a line, then its DCs are:
  l = ± a/√(a²+b²+c²), m = ± b/√(a²+b²+c²), n = ± c/√(a²+b²+c²)
  (The sign depends on the direction of the line).
• DRs of a Line Joining Two Points: The DRs of the line segment joining P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) are:
  (x₂ - x₁), (y₂ - y₁), (z₂ - z₁)

3. Equation of a Line in Space

• Vector Form:
  • Equation of a line passing through a point A (with position vector a) and parallel to a vector b:
    r = a + λb (where r is the position vector of any point on the line, λ is a scalar parameter)
• Cartesian Form:
  • Equation of a line passing through the point (x₁, y₁, z₁) and having direction ratios a, b, c:
    (x - x₁)/a = (y - y₁)/b = (z - z₁)/c
• Line Passing Through Two Points:
  • Vector Form: Equation of a line passing through two points A and B with position vectors a and b:
    r = a + λ(b - a)
  • Cartesian Form: Equation of a line passing through points (x₁, y₁, z₁) and (x₂, y₂, z₂):
    (x - x₁)/(x₂ - x₁) = (y - y₁)/(y₂ - y₁) = (z - z₁)/(z₂ - z₁)

4. Angle Between Two Lines

• Let the two lines be:
  • L₁: r = a₁ + λb₁
  • L₂: r = a₂ + μb₂
  • Or in Cartesian form with DRs (a₁, b₁, c₁) and (a₂, b₂, c₂).
• The angle θ between the lines is the angle between their parallel vectors b₁ and b₂.
• Vector Form:
  cos θ = |(b₁ ⋅ b₂)| / (|b₁| |b₂|)
• Cartesian Form:
  cos θ = |a₁a₂ + b₁b₂ + c₁c₂| / [ √(a₁²+b₁²+c₁²) * √(a₂²+b₂²+c₂²) ]
• Conditions:
  • Perpendicular Lines: θ = 90° ⇒ cos θ = 0
    ⇒ b₁ ⋅ b₂ = 0 or a₁a₂ + b₁b₂ + c₁c₂ = 0
  • Parallel Lines: θ = 0° ⇒ cos θ = 1
    ⇒ b₁ is parallel to b₂ ( b₁ = kb₂ for some scalar k)
    ⇒ a₁/a₂ = b₁/b₂ = c₁/c₂

5. Shortest Distance Between Two Lines

• Skew Lines: Lines that are neither parallel nor intersecting.
  • Lines: r = a₁ + λb₁ and r = a₂ + μb₂
  • Vector Form: Shortest Distance (SD) = | ((a₂ - a₁) ⋅ (b₁ × b₂)) | / |b₁ × b₂|
  • Cartesian Form: (Generally more complex, often easier to convert to vector form).
    SD = | det | / √[ (b₁c₂-b₂c₁)² + (c₁a₂-c₂a₁)² + (a₁b₂-a₂b₁)² ]
    where det = | (x₂-x₁) (y₂-y₁) (z₂-z₁) |
    | a₁ b₁ c₁ |
    | a₂ b₂ c₂ |
  • If SD = 0, the lines are intersecting. Condition for intersection: (a₂ - a₁) ⋅ (b₁ × b₂) = 0.
• Parallel Lines: Lines with parallel direction vectors (b₁ = b and b₂ = b).
  • Lines: r = a₁ + λb and r = a₂ + μb
  • Distance = | ((a₂ - a₁) × b) | / |b|

6. Equation of a Plane

• Normal Form:
  • Vector Form: r ⋅ n̂ = d
    (where n̂ is the unit vector normal to the plane from the origin, and d is the perpendicular distance from the origin to the plane).
  • Cartesian Form: lx + my + nz = d
    (where l, m, n are the DCs of the normal to the plane, and d is the distance from the origin).
• General Form (Point and Normal):
  • Vector Form: Equation of a plane passing through a point A (position vector a) and perpendicular to vector N:
    (r - a) ⋅ N = 0 or r ⋅ N = a ⋅ N
  • Cartesian Form: Equation of a plane passing through (x₁, y₁, z₁) and having DRs of the normal as A, B, C:
    A(x - x₁) + B(y - y₁) + C(z - z₁) = 0
    The general equation of a plane is Ax + By + Cz + D = 0, where A, B, C are DRs of the normal.
• Intercept Form: Equation of a plane making intercepts a, b, c on the x, y, z axes respectively:
  x/a + y/b + z/c = 1
• Plane Through Three Non-collinear Points:
  • Let the points be A(a), B(b), C(c).
  • Vector Form: (r - a) ⋅ [ (b - a) × (c - a) ] = 0
  • Cartesian Form: Let points be (x₁, y₁, z₁), (x₂, y₂, z₂), (x₃, y₃, z₃).
    | (x - x₁) (y - y₁) (z - z₁) |
    | (x₂ - x₁) (y₂ - y₁) (z₂ - z₁) | = 0
    | (x₃ - x₁) (y₃ - y₁) (z₃ - z₁) |

7. Coplanarity of Two Lines

• Two lines r = a₁ + λb₁ and r = a₂ + μb₂ are coplanar if the vector connecting a point on the first line to a point on the second line (a₂ - a₁) is perpendicular to the vector normal to both lines (b₁ × b₂).
• Condition for Coplanarity (Vector Form):
  (a₂ - a₁) ⋅ (b₁ × b₂) = 0
• Condition for Coplanarity (Cartesian Form):
  Lines: (x-x₁)/a₁ = ... and (x-x₂)/a₂ = ...
  | (x₂ - x₁) (y₂ - y₁) (z₂ - z₁) |
  | a₁ b₁ c₁ | = 0
  | a₂ b₂ c₂ |

8. Angle Between Two Planes

• The angle between two planes is defined as the angle between their normals.
• Let the planes be:
  • r ⋅ N₁ = d₁ and r ⋅ N₂ = d₂
  • A₁x + B₁y + C₁z + D₁ = 0 and A₂x + B₂y + C₂z + D₂ = 0
• Vector Form: If θ is the angle,
  cos θ = |(N₁ ⋅ N₂)| / (|N₁| |N₂|)
• Cartesian Form:
  cos θ = |A₁A₂ + B₁B₂ + C₁C₂| / [ √(A₁²+B₁²+C₁²) * √(A₂²+B₂²+C₂²) ]
• Conditions:
  • Perpendicular Planes: θ = 90° ⇒ cos θ = 0
    ⇒ N₁ ⋅ N₂ = 0 or A₁A₂ + B₁B₂ + C₁C₂ = 0
  • Parallel Planes: θ = 0° ⇒ cos θ = 1
    ⇒ N₁ is parallel to N₂ (N₁ = kN₂)
    ⇒ A₁/A₂ = B₁/B₂ = C₁/C₂

9. Distance of a Point from a Plane

• Vector Form: Distance from point P (position vector a) to the plane r ⋅ N = D:
  Distance = | (a ⋅ N - D) | / |N|
• Cartesian Form: Distance from point (x₁, y₁, z₁) to the plane Ax + By + Cz + D = 0:
  Distance = | Ax₁ + By₁ + Cz₁ + D | / √(A² + B² + C²)
• Distance from Origin: Set (x₁, y₁, z₁) = (0,0,0)
  Distance = | D | / √(A² + B² + C²)

10. Angle Between a Line and a Plane

• Let the line be r = a + λb (DRs a, b, c) and the plane be r ⋅ N = d (or Ax + By + Cz + D = 0).
• Let θ be the angle between the line and the plane. Let φ be the angle between the line and the normal to the plane. Then θ + φ = 90°.
• We find φ using the dot product of b and N.
  cos φ = |(b ⋅ N)| / (|b| |N|)
• Therefore, sin θ = cos φ.
• Vector Form:
  sin θ = |(b ⋅ N)| / (|b| |N|)
• Cartesian Form:
  sin θ = | Aa + Bb + Cc | / [ √(A² + B² + C²) * √(a² + b² + c²) ]
• Conditions:
  • Line Parallel to Plane: θ = 0° ⇒ sin θ = 0
    ⇒ b ⋅ N = 0 or Aa + Bb + Cc = 0 (Line is perpendicular to the normal)
  • Line Perpendicular to Plane: θ = 90° ⇒ sin θ = 1
    ⇒ b is parallel to N (b = kN)
    ⇒ a/A = b/B = c/C (Line is parallel to the normal)

Practice Multiple Choice Questions (MCQs)

1. If a line makes angles 90°, 135°, 45° with the x, y, and z axes respectively, its direction cosines are:
   a) 0, -1/√2, 1/√2
   b) 0, 1/√2, 1/√2
   c) 1, -1/√2, 1/√2
   d) 0, -1/√2, -1/√2

2. The distance of the point (2, 3, 4) from the plane 3x - 6y + 2z + 11 = 0 is:
   a) 1
   b) 2
   c) 7/7
   d) 1/7

3. The direction ratios of the line joining points (1, 0, -2) and (3, 2, -1) are:
   a) 2, 2, 1
   b) 4, 2, -3
   c) 2, 2, -1
   d) -2, -2, -1

4. The vector equation of the line passing through the point (5, 2, -4) and parallel to the vector 3i + 2j - 8k is:
   a) r = (3i + 2j - 8k) + λ(5i + 2j - 4k)
   b) r = (5i + 2j - 4k) + λ(3i + 2j - 8k)
   c) r = (5i + 2j - 4k) + λ(3i - 2j + 8k)
   d) r = (3i + 2j - 8k) + λ(5i - 2j + 4k)

5. The angle between the lines (x-1)/2 = (y-2)/3 = (z-3)/6 and x/1 = y/2 = z/2 is:
   a) cos⁻¹(20/21)
   b) cos⁻¹(8/21)
   c) sin⁻¹(8/21)
   d) 90°

6. The equation of the plane with intercepts 2, 3, and 4 on the x, y, and z axes respectively is:
   a) 2x + 3y + 4z = 1
   b) x/2 + y/3 + z/4 = 1
   c) 2x + 3y + 4z = 12
   d) x/2 + y/3 + z/4 = 12

7. If the lines (x-1)/(-3) = (y-2)/(2k) = (z-3)/2 and (x-1)/(3k) = (y-1)/1 = (z-6)/(-5) are perpendicular, then the value of k is:
   a) -10/7
   b) 10/7
   c) -7/10
   d) 7/10

8. The distance of the plane 2x - 3y + 6z + 14 = 0 from the origin is:
   a) 14
   b) 2
   c) 7
   d) 14/7

9. The reflection of the point (α, β, γ) in the XY-plane is:
   a) (α, β, 0)
   b) (0, 0, γ)
   c) (-α, -β, γ)
   d) (α, β, -γ)

10. The angle between the planes 2x - y + z = 6 and x + y + 2z = 7 is:
    a) 30°
    b) 45°
    c) 60°
    d) 90°

Answer Key:

1. a) (cos 90°=0, cos 135°=-1/√2, cos 45°=1/√2)
2. d) Distance = |3(2) - 6(3) + 2(4) + 11| / √(3² + (-6)² + 2²) = |6 - 18 + 8 + 11| / √(9 + 36 + 4) = |7| / √49 = 7/7 = 1. Rechecking calculation: |6 - 18 + 8 + 11| = |-12 + 8 + 11| = |-4 + 11| = 7. Denominator = sqrt(9+36+4) = sqrt(49) = 7. Distance = 7/7 = 1. Let me recheck the options. Ah, option (a) is 1. Option (d) is 1/7. The calculation gives 1. So the answer should be (a). Let me correct the key.
3. a) (3-1, 2-0, -1-(-2)) = (2, 2, 1)
4. b) r = a + λb, where a = 5i + 2j - 4k and b = 3i + 2j - 8k
5. a) DRs are (2, 3, 6) and (1, 2, 2). |b₁|=√(4+9+36)=√49=7. |b₂|=√(1+4+4)=√9=3. b₁⋅b₂ = 2(1)+3(2)+6(2) = 2+6+12=20. cos θ = |20| / (7 * 3) = 20/21. θ = cos⁻¹(20/21).
6. b) Using intercept form x/a + y/b + z/c = 1.
7. a) Condition for perpendicularity: a₁a₂ + b₁b₂ + c₁c₂ = 0. (-3)(3k) + (2k)(1) + (2)(-5) = 0 ⇒ -9k + 2k - 10 = 0 ⇒ -7k = 10 ⇒ k = -10/7.
8. b) Distance = |D| / √(A²+B²+C²) = |14| / √(2² + (-3)² + 6²) = 14 / √(4 + 9 + 36) = 14 / √49 = 14 / 7 = 2.
9. d) In reflection across the XY-plane, only the z-coordinate changes its sign.
10. c) Normals are N₁ = 2i - j + k and N₂ = i + j + 2k. |N₁|=√(4+1+1)=√6. |N₂|=√(1+1+4)=√6. N₁⋅N₂ = 2(1)+(-1)(1)+1(2) = 2-1+2=3. cos θ = |3| / (√6 * √6) = 3/6 = 1/2. θ = cos⁻¹(1/2) = 60°.

Corrected Answer Key:

1. a
2. a (Calculation yields 1)
3. a
4. b
5. a
6. b
7. a
8. b
9. d
10. c

Make sure you understand the derivation of these formulae and practice applying them to various problems. This chapter requires visualization and careful application of vector algebra. Good luck with your preparation!