Types of Vectors: Definition, Types, Properties, Applications & Class 12 Maths Notes

 In NCERT, a vector is defined as a quantity that has both magnitude and direction. A vector is used to find the displacement, velocity, acceleration, force, momentum, weight, electric field intensity, etc. It is denoted by an arrow. The arrow indicates the magnitude, and the arrowhead the direction. For example: →AB.

Vectors are classified into various types based on their properties and relationships with other vectors. Different types of vectors help to perform various arithmetic calculations.

• Zero Vector: A vector that has zero (0) magnitude is called a zero vector. 
• Unit Vectors: A vector is described as a unit vector if its magnitude is equal to 1. A unit vector is denoted as â = a / |a|.  Examples: î, ĵ, k̂ on x, y, z axes. 
• Coinitial Vectors: When two or more vectors have the same initial point then they are called coinitial vectors.

• Collinear Vectors: If two or more vectors are parallel to the same line, irrespective of their magnitude and direction they are called collinear vectors. It is also known as parallel vector. 

• Equal Vector: When two vectors have the same magnitude and direction, whatever their initial position is, they are called equal vectors. 

• Negative of a Vector: If two vectors have same magnitude but opposite in direction, then it is referred as negative vector.  

 

Vectors can perform various operations such as adding, subtracting, multiplying etc. Below are the operations of vectors. 

1. Addition of Vectors

• To perform addition in vector we use Triangle law of vector addition and parallelogram law of vector addition.

• Vector addition is commutative and associative.

2. Subtraction of Vectors

• If you want to subtract two vectors, then add its negative.

3. Multiplication of Vectors

• In vector multiplication is performed in two ways dot product and cross product.

• Dot product of two vector is scalar, and the cross product is vector.

Important Topics:

NCERT Class 11 notes	CBSE NCERT Class 11 Physics
Class 11 Chemistry notes	Class 11 Maths notes

 We know that vectors are not limited to mathematics and physics. It is also used in engineering and many real-life situations. Below are a few important applications of vectors. 

1. Physics

• In physics we use vectors to find force, acceleration and displacement. For example: If two or more forces are acting on a body, then what is the resultant force. 

2. Engineering and Mechanics

• Vectors are important in making machines, buildings and bridges because they help to identify the stresses, strain and forces acting on them. 

3. Navigation and GPS

• When travelling we use navigation and GPS system to determine the direction and distance. These systems use vectors for accurate data. 

4. Animations

• In video games the movement, 3D models and realistic animations are created using the vector. 

5. Electromagnetism

• In electromagnetism, vectors are used to represent the electric and magnetic field. This makes it easy to understand the flow of current and wave propagation. 

Also Read: NCERT Solution for class 11 & 12

Types of Vectors is present in class XII mathematics chapter Vector Algebra. The chapter's applications also have its importance in other subjects like physics. In class XII mathematics exams the weightage of vector algebra along with three-dimensional geometry ranges from 12 to 14 marks. 

The chapter of Vector Algebra also includes topics such as:

• Operations of Vectors(Addition, Subtraction)
• A Scalar multiplication with a Vector
• 2 Vectors' Product

1. From the given figure, identify which vectors are Collinear, Coinitial, and Equal Vectors.

Solution.

2. If the given vectors   are equal vectors then evaluate the values of a, b, and c.

Solution.

If the 2 given vectors are equal then 

a = 2, 

b = 2, 

and c = 1.

3.