This chapter focuses on the concepts and methods related to three-dimensional geometry, essential for understanding spatial relationships in mathematics.
Three Dimensional Geometry – Formula & Equation Sheet
Essential formulas and equations from Mathematics Part - II, tailored for Class 12 in Mathematics.
This one-pager compiles key formulas and equations from the Three Dimensional Geometry chapter of Mathematics Part - II. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Key concepts & formulas
Essential formulas, key terms, and important concepts for quick reference and revision.
Formulas
l = a / √(a² + b² + c²)
l is the direction cosine along x-axis and a, b, c are direction ratios. This formula helps express the direction cosine of a line in terms of its direction ratios.
m = b / √(a² + b² + c²)
m represents the direction cosine along y-axis. This illustrates how direction cosines relate to direction ratios through their magnitude.
n = c / √(a² + b² + c²)
n is the direction cosine along z-axis. Understanding this helps in deriving direction cosines from proportional direction ratios.
l² + m² + n² = 1
This equation indicates that the sum of squares of the direction cosines of a line is always equal to 1, confirming their geometric interpretation.
PQ = √((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²)
PQ is the distance between two points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂). A fundamental formula for calculating the distance in three-dimensional space.
r = a + λb
This vector equation represents a line in space, where r is the position vector of any point on the line, a is a point on the line, b is the direction vector, and λ is a scalar parameter.
(x - x₁) / a = (y - y₁) / b = (z - z₁) / c
This is the Cartesian form of the line equation that passes through point (x₁, y₁, z₁) and in the direction of vector (a, b, c). It is useful for solving geometric problems.
cos θ = (a₁a₂ + b₁b₂ + c₁c₂) / (√(a₁² + b₁² + c₁²)√(a₂² + b₂² + c₂²))
This formula calculates the angle θ between two lines with direction ratios (a₁, b₁, c₁) and (a₂, b₂, c₂). Useful in determining angular relationships.
d = |(a₂-a₁)・(b₁ × b₂)| / |b₁ × b₂|
This formula gives the shortest distance d between two skew lines defined by direction vectors b₁ and b₂ and position vectors a₁ and a₂.
pk = a + λb
This equation helps to define the position of line segments in terms of parameters. Identifying the relationship between points and direction vectors is key in geometry.
Equations
x = x₁ + λa
The parametric equation representing the x-coordinate of a point on a line in space, defined by point (x₁, y₁, z₁) with direction ratio a.
y = y₁ + λb
Similar to the x-equation, this represents the y-coordinate based on direction ratios when given starting point coordinates.
z = z₁ + λc
This connects the z-coordinate of the point on the directed line to the parameter λ and direction ratio c.
cos θ = |(a₁・a₂)| / (|a₁| |a₂|)
This represents the cos θ for two vectors with given direction ratios, demonstrating the geometric relationship between their angles.
PQ = (x₂ - x₁, y₂ - y₁, z₂ - z₁)
This vector representation shows the directional segment joining two points in space, essential in analyzing spatial relationships.
d = ||(PQ × (b₁ × b₂))|| / ||(b₁ × b₂)||
The cross product is employed here to comprehend the spatial positioning of lines in three dimensions.
x₁ + λa = x₂ + μb
This equation equates two lines at point coordinates based on parameters λ and μ, symbolizing intersection conditions.
l₁・l₂ = 0
When the dot product equals zero, this expresses perpendicularity between two direction vectors l₁ and l₂.
b₁ / b₁ = b₂ / b₂ = b₃ / b₃
This indicates the need for proportional direction ratios when comparing lines, clarifying relationships in geometry.
d = |a * b| / |b|
This gives the distance between a point and a line in vector form, essential in establishing relationships among geometric figures.
This chapter covers the concept of integrals, including indefinite and definite integrals, crucial for calculating areas under curves and solving practical problems in various fields.
Start chapterThis chapter explores how to use integrals to find areas under curves, between lines, and enclosed by shapes like circles and parabolas. Understanding these applications is crucial for solving real-world problems.
Start chapterThis chapter introduces differential equations, including their types and applications across various scientific fields.
Start chapterThis chapter introduces the fundamental concepts of vectors and their operations, which are crucial in mathematics, physics, and engineering.
Start chapterThis chapter focuses on linear programming, a method used to optimize certain objectives within given constraints, which is applicable in various fields like economics and management.
Start chapterThis chapter introduces the fundamental concepts of probability, including conditional probability and its applications which are essential for understanding uncertainty in random experiments.
Start chapter
