Topics
Mathematical Logic
- Statements and Truth Values in Mathematical Logic
- Logical Connectives
- Tautology, Contradiction, and Contingency
- Quantifier, Quantified and Duality Statements in Logic
- Negations of Compound Statements
- Converse, Inverse, and Contrapositive
- Algebra of Statements
- Application of Logic to Switching Circuits
- Overview of Mathematical Logic
Matrices
- Elementry Transformations
- Adjoint & Inverse of Matrix
- Application of Matrices
- Overview of Matrices
Trigonometric Functions
- Trigonometric Equations and Their Solutions
- Solutions of Triangle>Polar Co-Ordinates
- Solving a Triangle>Solving a Triangle
- Basics of Inverse Trigonometric Functions
- Graphs of Inverse Trigonometric Functions
- Domain, Range & Principal Value
- Properties of Inverse Trigonometric Functions
- Overview of Trigonometric Functions
Pair of Straight Lines
Vectors
- Overview of Vectors
- Basic Concepts of Vector Algebra
- Types of Vectors in Algebra
- Algebra of Vectors > Scalar Multiplication
- Algebra of Vectors > Addition & Subtraction of Two Vectors
- Collinearity and Coplanarity of Vectors
- Vectors in Coordinate Geometry
- Components of Vector in Algebra
- Vector Joining Two Points in Algebra
- Section Formula in Vector Algebra
- Product of Two Vectors > Scalar (Dot) Product
- Product of Two Vectors > Vector (Cross) Product
- Direction Ratios, Direction Cosine & Direction Angles in Vector
- Scalar Triple Product
- Vector Triple Product
Line and Plane
Linear Programming
Differentiation
- Introduction & Derivatives of Some Standard Functions
- Derivatives of Composite Functions
- Geometrical Meaning of Derivative
- Derivative of Inverse Function
- Logarithmic Differentiation
- Derivative of Implicit Functions
- Derivatives of Functions in Parametric Forms
- Higher Order Derivatives
- Overview of Differentiation
Applications of Derivatives
- Applications of Derivatives in Geometry
- Derivatives as a Rate Measure
- Approximations
- Rolle's Theorem
- Lagrange's Mean Value Theorem (LMVT)
- Increasing and Decreasing Functions
- Maxima and Minima
- Overview of Applications of Derivatives
Indefinite Integration
Definite Integration
- Definite Integral as Limit of Sum
- Integral Calculus
- Methods of Evaluation and Properties of Definite Integral
- Overview of Definite Integration
Application of Definite Integration
- Application of Definite Integration
- Area Bounded by Two Curves
- Overview of Application of Definite Integration
Differential Equations
- Basic Concepts of Differential Equations
- Order and Degree of a Differential Equation
- Formation of Differential Equations
- Methods of Solving Differential Equations> Homogeneous Differential Equations
- Methods of Solving Differential Equations>Linear Differential Equations
- Applications of Differential Equation
- Solution of a Differential Equation
- Overview of Differential Equations
Probability Distributions
- Random Variables
- Probability Distribution of Discrete Random Variables
- Probability Distribution of a Continuous Random Variable
- Variance of a Random Variable
- Expected Value and Variance of a Random Variable
- Overview of Probability Distributions
Binomial Distribution
Estimated time: 7 minutes
CBSE: Class 12
Introduction
A vector in three-dimensional geometry can be written in terms of its components along the x-, y-, and z-axes using the unit vectors \[\vec{i}, \vec{j}, \vec{k}\]. This component form makes it easier to find magnitude, compare vectors, and perform operations like addition and subtraction.
CBSE: Class 12
Definition: Component Form of a Vector
If P(x, y, z) is a point, then its position vector is
\[\vec{OP} = x\hat{i} + y\hat{j} + z\hat{k}\]
This is called the component form of a vector.
CBSE: Class 12
Properties of Vectors in Component Form
- Magnitude
\[|\vec{r}| = \sqrt{x^2 + y^2 + z^2}\]
- Addition
If \[\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}\] and
\[\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}\]
then \[\vec{a} + \vec{b} = (a_1 + b_1)\hat{i} + (a_2 + b_2)\hat{j} + (a_3 + b_3)\hat{k}\]
- Subtraction
\[\vec{a} - \vec{b} = (a_1 - b_1)\hat{i} + (a_2 - b_2)\hat{j} + (a_3 - b_3)\hat{k}\]
- Scalar Multiplication
\[\lambda\vec{a} = (\lambda a_1)\hat{i} + (\lambda a_2)\hat{j} + (\lambda a_3)\hat{k}\]
- Equality of Vectors
Two vectors are equal if their corresponding components are equal.
- Collinearity of Vectors
Two vectors are collinear if one is a scalar multiple of the other, i.e. \[\vec{b} = \lambda\vec{a}\]
- Equality of Vectors
Two vectors are equal if their corresponding components are equal.
CBSE: Class 12
Example 1
Find the values of \[x, y\] and \[z\] so that the vectors \[\vec{a} = x\hat{i} + 2\hat{j} + z\hat{k}\]and \[\vec{b} = 2\hat{i} + y\hat{j} + \hat{k}\]are equal.
Solution: Note that two vectors are equal if and only if their corresponding components are equal. Thus, the given vectors \[\vec{a}\] and \[\vec{b}\] will be equal if and only if
\[x = 2, \quad y = 2, \quad z = 1\]
CBSE: Class 12
Example 2
Question: Find a vector of magnitude 7 units in the direction of:
\[\vec{a} = \vec{i} - 2\vec{j}\]
Solution:
Step 1: Find the magnitude of \[\vec{a}\].
\[|\vec{a}| = \sqrt{1^2 + (-2)^2} = \sqrt{5}\]
Step 2: Find the unit vector in the same direction.
\[\hat{a} = \frac{1}{\sqrt{5}}(\vec{i} - 2\vec{j})\]
Step 3: Multiply by 7.
\[\vec{v} = 7\hat{a} = \frac{7}{\sqrt{5}}(\vec{i} - 2\vec{j})\]
\[\vec{v} = \frac{7}{\sqrt{5}}\vec{i} - \frac{14}{\sqrt{5}}\vec{j}\]
Video Tutorials
Shaalaa.com | Components of Vector
to track your progress
