Topics
Relations and Functions
Relations and Functions
Inverse Trigonometric Functions
Algebra
Matrices
- Concept of Matrices
- Types of Matrices
- Equality of Matrices
- Operation on Matrices
- Multiplication of a Matrix by a Scalar
- Properties of Matrix Addition
- Properties of Matrix Multiplication
- Transpose of a Matrix
- Symmetric and Skew Symmetric Matrices
- Invertible Matrices
- Overview of Matrices
- Negative of Matrix
- Operation on Matrices
- Proof of the Uniqueness of Inverse
- Elementary Transformations
Calculus
Vectors and Three-dimensional Geometry
Determinants
- Determinants of Matrix of Order One and Two
- Determinant of a Matrix of Order 3 × 3
- Minors and Co-factors
- Inverse of a Square Matrix by the Adjoint Method
- Applications of Determinants and Matrices
- Elementary Transformations
- Properties of Determinants
- Determinant of a Square Matrix
- Rule A=KB
- Overview of Determinants
- Geometric Interpretation of the Area of a Triangle
Linear Programming
Continuity and Differentiability
- Concept of Continuity
- Algebra of Continuous Functions
- Concept of Differentiability
- Derivatives of Composite Functions - Chain Rule
- Derivatives of Implicit Functions
- Derivatives of Inverse Trigonometric Functions
- Exponential and Logarithmic Functions
- Logarithmic Differentiation
- Derivatives of Functions in Parametric Forms
- Second Order Derivative
- Derivative - Exponential and Log
- Proof Derivative X^n Sin Cos Tan
- Infinite Series
- Higher Order Derivative
- Continuous Function of Point
- Mean Value Theorem
- Overview of Continuity and Differentiability
Applications of Derivatives
- Introduction to Applications of Derivatives
- Rate of Change of Bodies or Quantities
- Increasing and Decreasing Functions
- Maxima and Minima
- Maximum and Minimum Values of a Function in a Closed Interval
- Simple Problems on Applications of Derivatives
- Graph of Maxima and Minima
- Approximations
- Tangents and Normals
- Overview of Applications of Derivatives
Probability
Sets
Integrals
- Introduction of Integrals
- Integration as an Inverse Process of Differentiation
- Some Properties of Indefinite Integral
- Methods of Integration: Integration by Substitution
- Integration Using Trigonometric Identities
- Integrals of Some Particular Functions
- Methods of Integration: Integration Using Partial Fractions
- Methods of Integration: Integration by Parts
- Fundamental Theorem of Calculus
- Evaluation of Definite Integrals by Substitution
- Properties of Definite Integrals
- Definite Integrals
- Indefinite Integral Problems
- Comparison Between Differentiation and Integration
- Geometrical Interpretation of Indefinite Integrals
- Indefinite Integral by Inspection
- Definite Integral as the Limit of a Sum
- Evaluation of Simple Integrals of the Following Types and Problems
- Overview of Integrals
Applications of the Integrals
Differential Equations
- Differential Equations
- Order and Degree of a Differential Equation
- General and Particular Solutions of a Differential Equation
- Linear Differential Equations
- Homogeneous Differential Equations
- Solutions of Linear Differential Equation
- Differential Equations with Variables Separable Method
- Formation of a Differential Equation Whose General Solution is Given
- Procedure to Form a Differential Equation that Will Represent a Given Family of Curves
- Overview of Differential Equations
Vectors
- Vector
- Basic Concepts of Vector Algebra
- Direction Cosines
- Vector Operations>Addition and Subtraction of Vectors
- Properties of Vector Addition
- Vector Operations>Multiplication of a Vector by a Scalar
- Components of Vector
- Vector Joining Two Points
- Section Formula
- Vector (Or Cross) Product of Two Vectors
- Scalar (Or Dot) Product of Two Vectors
- Projection of a Vector on a Line
- Geometrical Interpretation of Scalar
- Scalar Triple Product of Vectors
- 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
- Forms of the Equation of a Straight Line
- 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
- Distance of a Point from a Plane
- Plane Passing Through the Intersection of Two Given Planes
- Overview of Three Dimensional Geometry
Linear Programming
Probability
- Direction cosines of a line passing through two points.
Notes
A directed line L passing through the origin makes angles α, β and γ with x, y and z-axes, respectively, called direction angles, then cosine of these angles, namely, cos α, cos β and cos γ are called direction cosines of the directed line L.
If we reverse the direction of L, then the direction angles are replaced by their supplements, i.e., π - α , π-β and π-γ . Thus, the signs of the direction cosines are reversed.in explain in following fig.
A given line in space can be extended in two opposite directions and so it has two sets of direction cosines. In order to have a unique set of direction cosines for a given line in space, we must take the given line as a directed line. These unique direction cosines are denoted by l, m and n.
Remark: If the given line in space does not pass through the origin, then, in order to find its direction cosines, we draw a line through the origin and parallel to the given line. Now take one of the directed lines from the origin and find its direction cosines as two parallel line have same set of direction cosines.
Any three numbers which are proportional to the direction cosines of a line are called the direction ratios of the line. If l, m, n are direction cosines and a, b, c are direction ratios of a line, then a = λl, b=λm and c = λn, for any nonzero λ ∈ R.
Let a, b, c be direction ratios of a line and let l, m and n be the direction cosines (d.c’s) of the line. Then
`l/a = m/b = n/c = k ("say") ` k being a constant.
Therefore l = ak , m=bk , n= ck ...(1)
But `l^2 + m^2 + n^2 = 1`
Therefore `k^2(a^2 + b^2 + c^2) = 1`
Or k = `+- 1/ sqrt(a^2 + b^2 + c^2)`
Hence, from (1), the d.c.’s of the line are
`l = +- a/sqrt(a^2 + b^2 + c^2) ` ,
`m = +- b/sqrt(a^2 + b^2 + c^2)` ,
`n = +- c/sqrt(a^2 + b^2 + c^2)`
where, depending on the desired sign of k, either a positive or a negative sign is to be taken for l, m and n.
For any line, if a, b, c are direction ratios of a line, then ka, kb, kc ; k ≠ 0 is also a set of direction ratios. So, any two sets of direction ratios of a line are also proportional. Also, for any line there are infinitely many sets of direction ratios.
Video link : https://youtu.be/zeZdiru6J7M
Direction cosines of a line passing through two points :
The direction cosines of a line passing through the given points `P(x_1, y_1, z_1)` and `Q(x_2, y_2, z_2)` as following fig. 
Let l, m, n be the direction cosines of the line PQ and let it makes angles α, β and γ with the x, y and z-axis, respectively.
Draw perpendiculars from P and Q to XY-plane to meet at R and S. Draw a perpendicular from P to QS to meet at N. Now, in right angle triangle PNQ, ∠PQN= γ in following fig.
Therefore, cos γ = `(NQ)/(PQ) = (z_2 - z_1)/(PQ)`
Similarly cos α = `(x_2 - x_1)/(PQ)` and cos β =`(y_2 - y_1)/(PQ) `
Hence, the direction cosines of the line segment joining the points `P(x_1, y_1, z_1)` and `Q(x_2, y_2, z_2) ` are
`(x_2 - x_1)/(PQ)` , `(y_2 - y_1)/(PQ) ` , `(z_2 - z_1)/(PQ)`
where PQ = `sqrt ((x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2)`
Video link : https://youtu.be/QVQvcM4JPx0
