Topics
Relations and Functions
Relations and Functions
Inverse Trigonometric Functions
- Basics of Inverse Trigonometric Functions
- Domain, Range & Principal Value
- Graphs of Inverse Trigonometric Functions
- Properties of Inverse Trigonometric Functions
- Overview of Inverse Trigonometric Functions
Algebra
Calculus
Matrices
Determinants
- Determinant of a Matrix
- Expansion of Determinant
- Area of Triangle using Determinant
- Minors and Co-factors
- Adjoint & Inverse of Matrix
- Applications of Determinants and Matrices
- Overview of Determinants
Vectors and Three-dimensional Geometry
Continuity and Differentiability
- Continuous and Discontinuous Functions
- Algebra of Continuous Functions
- Concept of Differentiability
- Derivative of Composite Functions
- Derivative of Implicit Functions
- Derivative of Inverse Function
- Exponential and Logarithmic Functions
- Logarithmic Differentiation
- Derivatives of Functions in Parametric Forms
- Second Order Derivative
- Overview of Continuity and Differentiability
Linear Programming
Applications of Derivatives
Probability
Integrals
- Introduction of Integrals
- Integration as an Inverse Process of Differentiation
- Properties of Indefinite Integral
- Methods of Integration> Integration by Substitution
- Methods of Integration>Integration Using Trigonometric Identities
- Methods of Integration> Integration Using Partial Fraction
- Methods of Integration> Integration by Parts
- Integrals of Some Particular Functions
- Definite Integrals
- Fundamental Theorem of Integral Calculus
- Evaluation of Definite Integrals by Substitution
- Properties of Definite Integrals
- Overview of Integrals
Sets
Applications of the Integrals
Differential Equations
- Basic Concepts of Differential Equations
- Order and Degree of a Differential Equation
- General and Particular Solutions of a Differential Equation
- Methods of Solving Differential Equations> Variable Separable Differential Equations
- Methods of Solving Differential Equations> Homogeneous Differential Equations
- Methods of Solving Differential Equations>Linear Differential Equations
- Overview of Differential Equations
Vectors
- Vector
- Basic Concepts of Vector Algebra
- Direction Ratios, Direction Cosine & Direction Angles
- Types of Vectors in Algebra
- Vector Operations>Addition and Subtraction of Vectors
- Algebra of Vector Addition
- Components of Vector in Algebra
- Vector Joining Two Points in Algebra
- Section Formula in Coordinate Geometry
- Vector (Or Cross) Product of Two Vectors
- Algebraic Products of Vectors
- Projection of a Vector on a Line
- Geometrical Interpretation of Scalar
- Scalar Triple Product
- Position Vector of a Point Dividing a Line Segment in a Given Ratio
- Magnitude and Direction of a Vector
- Vectors Examples and Solutions
- Introduction of Product of Two Vectors
- Overview of Vectors
Three - Dimensional Geometry
- Introduction of Three Dimensional Geometry
- Direction Cosines and Direction Ratios of a Line
- Relation Between Direction Ratio and Direction Cosines
- Equation of a Line in Space
- Angle Between Two Lines
- Shortest Distance Between Two Lines
- Three - Dimensional Geometry Examples and Solutions
- Equation of a Plane Passing Through Three Non Collinear Points
- Equations of Line in Different Forms
- Coplanarity of Two Lines
- Distance of a Point from a Plane
- Angle Between Line and a Plane
- Angle Between Two Planes
- Vector and Cartesian Equation of a Plane
- Equation of a Plane in Normal Form
- Equation of a Plane Perpendicular to a Given Vector and Passing Through a Given Point
- Plane Passing Through the Intersection of Two Given Planes
- Overview of Three Dimensional Geometry
Linear Programming
Probability
Notes
Let `π_1` and `π_2` be two planes with equations `vec r . hat n _1 = d_1` and `vec r . hat n _2 = d_2` respectively. The position vector of any point on the line of intersection must satisfy both the equations fig.
If `vec t ` is the position vector of a point on the line , then
`vec t . hat n_1 = d_1` and `vec t . hat n _2 = d_2`
Therefore , for all real values of λ, we have
`vec t . (hat n _1 + lambda hat n_2) = d_1 + lambda d_2`
Since `vec t` is arbitrary, it satisfies for any point on the line.
Hence , the equation `vec r . (vec n_1 + lambda vec n_2) = d_1 + lambda d_2` represents a plane `π_3` which is such that if any vector ` vec r` satisfies both the equations `π_1` and `π_2`, it also satisfies the equation `π_3` i.e., any plane passing through the intersection of the planes
`vec r . vec n_1 = d_1` and `vec r . vec n_2 = d_2`
has the equation `vec r . (vec n_1 + lambda vec n_2) = d_1 + lambda d_2` ...(1)
Cartesian form:
In Cartesian system, let `vec n_1 = A_1 hat i + B_2 hat j + C_1 hat k`
`vec n_2 = A_2 hat i + B_2 hat j + C _2 hat k`
and `vec r = x hat i + y hat j + z hat k`
Then (1) becomes
`x (A_1 + lambda A_2) + y(B_1 + lambda B_2) + z(C_1 +lambda C_2) = d_1 + lambda d_2`
or `(A_1x +B_1y + C_1z -d_1) + lambda (A_2x + B_2y + C_2z -d_2) = 0` ..(2)
which is the required Cartesian form of the equation of the plane passing through the intersection of the given planes for each value of λ.
