Topics
Relations and Functions
Relations and Functions
Inverse Trigonometric Functions
- Basics of Inverse Trigonometric Functions
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- Overview of Inverse Trigonometric Functions
Algebra
Calculus
Matrices
Determinants
- Determinant of a Matrix
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Vectors and Three-dimensional Geometry
Continuity and Differentiability
- Continuous and Discontinuous Functions
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Linear Programming
Applications of Derivatives
Probability
Integrals
- Introduction of Integrals
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- Properties of Indefinite Integral
- Methods of Integration> Integration by Substitution
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Sets
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Differential Equations
- Basic Concepts of Differential Equations
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- Methods of Solving Differential Equations> Variable Separable Differential Equations
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Vectors
- Vector
- Basic Concepts of Vector Algebra
- Direction Ratios, Direction Cosine & Direction Angles
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- 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
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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
If `P_1(x_1, y_1, z_1)` and `P_2(x_2, y_2, z_2)` are any two points, then the vector joining `P_1` and `P_2` is the vector in following fig.
Joining the points `P_1` and `P_2` with the origin O, and applying triangle law, from the triangle `OP_1P_2`, we have
`vec (OP_1) + vec (P_1P_2) = vec (OP_2)`
Using the properties of vector addition, the above equation becomes
`vec (P_1P_2) = vec (OP_2) - vec (OP_1)`
i.e. `vec (P_1P_2) = (x_2hat i + y_2hatj + z_2hat k) - (x_1hati + y_1hatj + z_1 hat k)`
= `(x_2 - x_1)hat i + (y_2 - y_1)hat j + (z_2 -z_1) hat k`
The magnitude of vector `vec(P_1P_2)` is given by
`|vec (P_1P_2)| = sqrt ((x_2 -x_1)^2 + (y_2 - y_1)^2 + (z_2-z_1)^2)`
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