CBSE Class 12 Maths Chapter 11 Three Dimensional Geometry Notes

Key Definitions in 3D Geometry

Direction Cosines (DCs)

The cosines of the angles (α, β, γ) made by a directed line with the positive direction of the x, y, and z-axes respectively. Denoted by l, m, n. We have l = cosα, m = cosβ, n = cosγ, and l² + m² + n² = 1.

Direction Ratios (DRs)

Any three numbers (a, b, c) which are proportional to the direction cosines (l, m, n) of a line are called its direction ratios. If a, b, c are DRs, then l = ±a/√(a²+b²+c²), m = ±b/√(a²+b²+c²), n = ±c/√(a²+b²+c²).

Skew Lines

Two lines in space that are neither parallel nor intersecting are called skew lines. They lie in different planes.

Coplanar Lines

Two lines that lie on the same plane. Intersecting lines are always coplanar, and parallel lines are always coplanar.

Normal Vector to a Plane

A vector that is perpendicular to every vector lying in the plane. It defines the orientation of the plane in space.

Shortest Distance

The perpendicular distance between two lines (skew or parallel). For skew lines, it's the length of the common perpendicular. For parallel lines, it's the perpendicular distance between any point on one line to the other line.

Understanding Direction Cosines and Direction Ratios

In three-dimensional geometry, direction cosines (DCs) and direction ratios (DRs) are fundamental concepts used to describe the direction of a line in space. The direction cosines of a line are essentially the cosines of the angles that the line makes with the positive x, y, and z-axes. These angles are conventionally denoted by α, β, and γ, respectively. Thus, the direction cosines are l = cosα, m = cosβ, and n = cosγ. A crucial property of direction cosines is that the sum of their squares is always equal to one: l² + m² + n² = 1.

Direction ratios (DRs), on the other hand, are any three numbers (say, a, b, c) that are proportional to the direction cosines (l, m, n) of the line. This means that l/a = m/b = n/c = k for some constant k. While DCs uniquely define the direction (up to sign), DRs only provide the ratio of the components along the axes. From DRs (a, b, c), you can find the DCs using the formulas l = ±a/√(a²+b²+c²), m = ±b/√(a²+b²+c²), n = ±c/√(a²+b²+c²). The choice of sign depends on the directed sense of the line. If a line passes through two points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂), its direction ratios can be given by (x₂ - x₁, y₂ - y₁, z₂ - z₁). Understanding this relationship is key to solving many problems involving lines in 3D space.

Key Formulas & Relationships for Lines and Planes

• Direction Ratios (DRs) of a line joining P(x₁, y₁, z₁) and Q(x₂, y₂, z₂): (x₂ - x₁, y₂ - y₁, z₂ - z₁)
• Equation of a line through a point A(x₁, y₁, z₁) and parallel to vector b = ai + bj + ck: - Vector form: r = a + λb - Cartesian form: (x - x₁)/a = (y - y₁)/b = (z - z₁)/c
• Equation of a line passing through two points A(x₁, y₁, z₁) and B(x₂, y₂, z₂): - Vector form: r = a + λ(b - a) - Cartesian form: (x - x₁)/(x₂ - x₁) = (y - y₁)/(y₂ - y₁) = (z - z₁)/(z₂ - z₁)
• Angle θ between two lines: - Cos θ = |b₁ ⋅ b₂| / (|b₁| |b₂|) (Vector form, if lines parallel to b₁, b₂) - Cos θ = |a₁a₂ + b₁b₂ + c₁c₂| / (√(a₁²+b₁²+c₁²)√(a₂²+b₂²+c₂²)) (Cartesian form, if DRs are (a₁,b₁,c₁) and (a₂,b₂,c₂))
• Condition for perpendicular lines: b₁ ⋅ b₂ = 0 or a₁a₂ + b₁b₂ + c₁c₂ = 0
• Condition for parallel lines: b₁ = λb₂ or a₁/a₂ = b₁/b₂ = c₁/c₂
• Shortest distance between two skew lines r = a₁ + λb₁ and r = a₂ + μb₂: - d = |(a₂ - a₁) ⋅ (b₁ × b₂)| / |b₁ × b₂|
• Shortest distance between two parallel lines r = a₁ + λb and r = a₂ + μb: - d = |b × (a₂ - a₁)| / |b|
• Equation of a plane in normal form: - Vector form: r ⋅ n̂ = d (where n̂ is unit normal vector, d is distance from origin) - Cartesian form: lx + my + nz = d
• Equation of a plane passing through a point (x₁, y₁, z₁) and perpendicular to vector n = Ai + Bj + Ck: - Vector form: (r - a) ⋅ n = 0 - Cartesian form: A(x - x₁) + B(y - y₁) + C(z - z₁) = 0
• Equation of a plane passing through three non-collinear points (x₁, y₁, z₁), (x₂, y₂, z₂), (x₃, y₃, z₃): - Using determinant: |(x-x₁), (y-y₁), (z-z₁); (x₂-x₁), (y₂-y₁), (z₂-z₁); (x₃-x₁), (y₃-y₁), (z₃-z₁)| = 0
• Equation of a plane passing through the intersection of two planes P₁=0 and P₂=0: P₁ + λP₂ = 0

Worked Examples for 3D Geometry

• {"example":"Find the direction cosines of the line joining points A(1, -2, 3) and B(3, 4, -1).","solution":"DRs of AB = (3-1, 4-(-2), -1-3) = (2, 6, -4).\nMagnitude = √(2² + 6² + (-4)²) = √(4 + 36 + 16) = √56 = 2√14.\nDCs are (2/(2√14), 6/(2√14), -4/(2√14)) = (1/√14, 3/√14, -2/√14)."}
• {"example":"Find the vector and Cartesian equations of the line passing through the point (5, 2, -4) and parallel to the vector 3i + 2j - 8k.","solution":"Given point a = 5i + 2j - 4k.\nParallel vector b = 3i + 2j - 8k.\nVector equation: r = (5i + 2j - 4k) + λ(3i + 2j - 8k).\nCartesian equation: (x - 5)/3 = (y - 2)/2 = (z - (-4))/-8 => (x - 5)/3 = (y - 2)/2 = (z + 4)/-8."}
• {"example":"Determine if the lines (x-1)/2 = (y-2)/3 = (z-3)/4 and (x-4)/5 = (y-1)/2 = z/1 are perpendicular.","solution":"DRs of L1: (a₁,b₁,c₁) = (2, 3, 4).\nDRs of L2: (a₂,b₂,c₂) = (5, 2, 1).\nCheck condition for perpendicularity: a₁a₂ + b₁b₂ + c₁c₂ = (2)(5) + (3)(2) + (4)(1) = 10 + 6 + 4 = 20.\nSince 20 ≠ 0, the lines are not perpendicular."}

Equations of a Plane in Different Forms

Understanding the different forms of the equation of a plane is crucial for solving problems in 3D geometry. A plane is a flat, two-dimensional surface that extends infinitely in three-dimensional space. Its orientation and position can be uniquely determined by various conditions.

1. Equation of a Plane in Normal Form: This is one of the most fundamental forms. If n̂ is a unit normal vector to the plane, and d is the perpendicular distance of the plane from the origin, then the vector equation of the plane is r ⋅ n̂ = d. In Cartesian form, if n̂ = li + mj + nk (where l, m, n are the direction cosines of the normal) then the equation becomes lx + my + nz = d.

1. Equation of a Plane Perpendicular to a Given Vector and Passing Through a Given Point: If a plane passes through a point A with position vector a, and is perpendicular to a vector n, then any point P with position vector r on the plane must satisfy the condition that the vector AP (r - a) is perpendicular to n. Thus, the vector equation is (r - a) ⋅ n = 0. In Cartesian form, if A = (x₁, y₁, z₁) and n = Ai + Bj + Ck, the equation is A(x - x₁) + B(y - y₁) + C(z - z₁) = 0.

1. Equation of a Plane Passing Through Three Non-Collinear Points: If three points (x₁, y₁, z₁), (x₂, y₂, z₂), (x₃, y₃, z₃) are not collinear, they uniquely define a plane. The Cartesian equation can be found using the determinant form: | (x-x₁) (y-y₁) (z-z₁) |

| (x₂-x₁) (y₂-y₁) (z₂-z₁) | = 0
| (x₃-x₁) (y₃-y₁) (z₃-z₁) |
Alternatively, one can find two vectors lying in the plane (e.g., (P₂-P₁) and (P₃-P₁)), calculate their cross product to get the normal vector n, and then use the point-normal form (r - a) ⋅ n = 0.

1. Equation of a Plane in Intercept Form: If a plane makes intercepts a, b, c with the x, y, and z-axes respectively, its Cartesian equation is x/a + y/b + z/c = 1. This form is useful when dealing with intercepts.

1. Equation of a Plane Passing Through the Intersection of Two Given Planes: If P₁ = A₁x + B₁y + C₁z + D₁ = 0 and P₂ = A₂x + B₂y + C₂z + D₂ = 0 are the equations of two planes, then the equation of a plane passing through their intersection is P₁ + λP₂ = 0, where λ is any real number. This is a powerful tool for finding specific planes that satisfy additional conditions.

Comparing Vector and Cartesian Forms

Exam Tip: Avoiding Common Traps in 3D Geometry

When solving problems in Three Dimensional Geometry, pay close attention to the details to avoid common mistakes:

• Skew vs. Intersecting Lines: Always test for intersection before calculating the shortest distance between skew lines. If (a₂ - a₁) ⋅ (b₁ × b₂) = 0, the lines are coplanar. If b₁ × b₂ ≠ 0, they intersect. Only if they are not parallel AND not intersecting, use the skew line formula.
• Unit Normal Vector for Plane: In the r ⋅ n̂ = d form, n̂ must be a unit vector. If you use a non-unit normal vector n, the equation becomes r ⋅ n = p, where p is not the distance from the origin, but d = p/|n|.
• Signs in Direction Cosines: Remember that l, m, n define a directed line. If a problem doesn't specify a direction, the direction cosines can be ±l, ±m, ±n. However, l² + m² + n² = 1 always holds.
• Vector Operations: Be careful with dot products and cross products. A scalar triple product (a₂ - a₁) ⋅ (b₁ × b₂) is a scalar quantity, while b × (a₂ - a₁) is a vector. Their magnitudes are used in distance formulas.

Practice Questions with Solutions

• Q: What is the relationship between direction cosines (l, m, n) of a line? A: l² + m² + n² = 1.
• Q: When are two lines in space considered 'skew'? A: Two lines are skew if they are neither parallel nor intersecting.
• Q: Write the Cartesian equation of a plane that cuts intercepts 2, -3, 4 on the x, y, z-axes respectively. A: x/2 + y/(-3) + z/4 = 1.
• Q: What is the condition for two planes with normal vectors n₁ and n₂ to be perpendicular? A: The dot product of their normal vectors must be zero: n₁ ⋅ n₂ = 0.

Frequently Asked Questions