[Maths Class Notes] on Types of Vector Pdf for Exam

To recall, in Mathematics a physical quantity that has both magnitude and a direction is said to be a vector. A vector is a geometrical entity that has magnitude and direction. In a line, the length of a line is a magnitude, and the arrow is its direction. The start point is its tail, and the endpoint is its head. An increase and decrease in temperature are the best examples of a vector, it has both magnitude and direction.

Representation of Vectors

Vectors are normally represented by a bold lowercase letter, such as a, or by an arrow over the letter. Vectors can also be defined by their initial and terminal points, which can indicate by an arrow above them. The initial point of a vector is also known as the tail, while the terminal or endpoint is known as the head. Vectors describe how an object moves from one location to the other.

Operations on Vectors

Some basic vector operations can be done geometrically without the use of a coordinate system. Addition, subtraction, and multiplication by a scalar are the vector operations. In addition, the dot product and the cross product are two methods for multiplying two vectors together. These are explained as given below:

10 Types of Vectors

Here we will be discussing different types of vectors. There are commonly 10 different types of vectors frequently used in maths. The 10 types of vectors are:

1. Zero vector

2. Unit Vector

3. Position Vector

4. Co-initial Vector

5. Like and Unlike Vectors

6. Coplanar Vector

7. Collinear Vector

8. Equal Vector

9. Displacement Vector

10. Negative of a Vector

Let Us Discuss Them in Detail

1. Zero Vector or Null Vector

A vector is said to be a Zero Vector when the magnitude of the vector is zero and the starting point and the endpoint of the vector is the same. For instance, PQ is a line segment the coordinates of the point P are the same as that of the point Q. A Zero vector is denoted by 0.  The zero vector doesn’t have any specific direction.

2. Unit Vector

A vector is said to be a unit vector when the magnitude of the vector is 1 unit in length. Suppose if x is a vector having a magnitude x then the unit vector is denoted by x̂ in the direction of the vector, and it has a magnitude equal to 1. But two-unit vectors cannot be equal as they might have different directions.

3. Travelling Position 

Point X taken in the plane is said to be a position vector. It simply denotes the position. Let OX be a point in a plane concerning its origin. Let us consider O is taken as reference origin and X is an arbitrary point in the plane then the vector is called the position vector of the point.

4. Co-Initial Vectors

A vector is said to be a co-initial vector when two or more vectors have the same starting point, for example, Vectors AB and AC are called co-initial vectors because they have the same starting point A.

5. Like and Unlike Vectors

The vectors having the same directions are said to be like vectors whereas vectors having opposite directions are said to be unlike vectors.

6. Coplanar Vectors

Three or more vectors lying in the same plane are known as coplanar vec

7. Collinear Vectors

Vectors that lie in the parallel line or the same line concerning their magnitude and direction are known to be collinear vectors, also known as parallel vectors.

8. Equal Vectors

Two vectors are said to be equal vectors when they have both direction and magnitude equal, even if they have different initial points.

9. Displacement Vector

The vector AB represents a displacement vector if a point is displaced from position A to B.

10. Negative of a Vector

Suppose a vector is given with the same magnitude and direction, now if any vector with the same magnitude but the opposite direction is given then this vector is said to be negative of that vector. Consider two vectors a and b, such that they have the same magnitude but opposite in direction then these vectors can be written as

a = – b

Let Us See Some Examples to Understand the Types of Vectors.

Solved Examples:

Example 1: 

Identify the vectors 

Solution:

Vectors a and c are unit vectors because they have magnitude 1.

Vector a and c are equal vectors because they have the same magnitude and same direction

Vectors b, c and d have the same initial point, so they are co-initial vectors.

Example 2:

Let us take the points P (1, 0, 0), Q (0, 1, 0) and R(0, 0, 1) on the x-axis, y-axis and z-axis, respectively. Identify the vectors.

Solution:  

OP =1; OQ =1; 0R =1.

Therefore, the vectors OP, OQ and OR, are called unit vectors.

Quiz time

Identify the vectors

Solution:

1. Like vectors

2. opposite vectors

3. Unlike vectors

4. Equal vectors