What are Vectors?
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:
Addition of Vectors
Subtraction of Vectors
Scalar Multiplication
Scalar Triple Product of Vectors
Multiplication of Vectors
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:
Zero vector
Unit Vector
Position Vector
Co-initial Vector
Like and Unlike Vectors
Coplanar Vector
Collinear Vector
Equal Vector
Displacement Vector
Negative of a Vector
Let Us Discuss Them in Detail
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.
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.
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.
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.
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.
Coplanar Vectors
Three or more vectors lying in the same plane are known as coplanar vec
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.
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.
Displacement Vector
The vector AB represents a displacement vector if a point is displaced from position A to B.
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:
Like vectors
opposite vectors
Unlike vectors
Equal vectors
FAQs on Types of Vector
1. What is the collinear vector?
Two vectors say a and b are given then they are said to be collinear vectors if they are parallel to each other. They can have different magnitudes or equal magnitude and their direction can be the same or even opposite.
Let us get more familiar to collinear vectors with an example:
Suppose we have a vector given
Then the collinear vectors are:
From the above picture it is clear that all three vectors have different magnitude and different directions but they are parallel to each other so we can say that all the three vectors are collinear vectors.
2. What is the zero vector?
The zero vector is a vector with its magnitude as 0. It is also known as the null vector. Consider this figure, where the vector O covers 0 magnitude means its starting point and the initial point is the same then it is said to be a zero vector or null vector. The dimension of zero vector are always zero i.e. O(0,0) in two dimensional, O(0,0,0) in three dimensional and so on.
For example, A person traveling from a point 1 km to the north direction and returning back 1 km in the south direction to the same point. Here we can see that the magnitude is 2km but the starting point and the endpoint are the same so the magnitude is zero and said to be null so it is said to be a zero vector.
3. What are the examples of Vector Quantities?
Vector quantities are physical quantities that are precisely defined by magnitude and direction. Both the direction of the medium's movement and the measurement of the scalar quantity is referred to as vector quantities. You may have noticed that scalar products are usually the products of two vector quantities. Physical quantities such as displacement, force, velocity, position, torque, and so on are examples of vector quantities. Vectors include displacement, acceleration, magnetic polarization, and angular velocity. Each of these quantities differs in a complete description of the quantity requires both direction and magnitude.
4. What is the difference between Vector and Scalars?
The prime distinction between scalars and vectors is that a scalar is a quantity that is unchanged by direction, whereas a vector is a physical quantity that has both magnitude and direction. Scalars are units with magnitude but no direction, whereas vectors are units with both magnitude and direction. Actual values with units of measurement are commonly referred to as scalars. The direction in which an object is traveling is indicated by vectors. Scalar quantities are all one-dimensional, while vector quantities might be one, two, or three-dimensional. Vectors are denoted by bold letters, such as V for velocity, or by placing an arrowhead over the letter. Scalars are denoted by simple alphabets, such as V for velocity. Because a scalar quantity has the same value regardless of direction, it cannot be resolved. The sine or cosine of the nearby angle can be used to resolve a vector quantity in any direction.
5. How can Vectors be applied in real life?
Vectors are employed in everyday life to depict the velocity of an aircraft when both the speed and direction of travel of the aircraft are known. Electromagnetic induction is caused by the interaction of electric and magnetic forces. In everyday life, vectors are used to locate people and objects. They are also used to define objects that are acting because of an external force. Other ways to make use of Vectors in real life is - Identifying the rain's direction and holding your umbrella in that direction. To move an object in a specified direction, we must apply the appropriate force in that direction. Identifying the rain's direction and holding your umbrella in that direction. Vectors are used in various wave propagations such as AC wave propagation, vibration propagation, sound propagation etc.
6. How are vectors useful in the field of Physics and Mathematics?
Vectors are extremely useful in physics and mathematics. They are used to represent the position, displacement, velocity, and acceleration of physical objects and quantities.
Vectors are extremely important in the study of partial differential equations and differential geometry.
Vectors are useful in physics because they may visibly represent the travelling position, displacement, velocity, and acceleration.
Vectors are used considerably in physics and engineering, particularly in the study of electromagnetic fields, gravitational fields, and fluid flow.
To calculate an object's velocity, one must first determine how quickly the displacement is changing and in which direction it is changing. As a result, it is well represented by a vector.
Vector algebra is used to measure angles and distances between panels in satellites, in the development of pipe networks in many sectors, and in computing angles and distances between beams and structures in civil engineering.
7. Where can we get the best notes and study material to study about Vector?
The importance of notes and sample papers in the examination cannot be understated. These notes are very useful for those who do not take notes when studying or revising. These study materials will make exam preparation easier and more structured for learners. Revision Notes and Sample Papers for Vectors can be found on Vedantu's website. These will be available for download in PDF format to students. These study resources are available for free download. Students can get good grades if they practice using these study resources.