Vectors
Definition
The vectors are defined as an object containing both magnitude and direction. Vector describes the movement of an object from one point to another. Vector math can be geometrically picturised by the directed line segment.
The magnitude of a vector is shown by vertical lines on both the sides of the given vector “|a|”. It represents the length of the vector. Mathematically, the magnitude of a vector is calculated by the help of “Pythagoras theorem,” i.e.
|a|= √(x2+y2)
Types of Vector
Unit Vector
A unit vector has a length (or magnitude) equal to one, which is basically used to show the direction of any vector. A unit vector is equal to the ratio of a vector and its magnitude. Symbolically, it is represented by a cap or hat (^).
Zero Vector
A vector with zero magnitudes is called a zero vector. The coordinates of zero vector are given by (0,0,0) and it is usually represented by 0 with an arrow (→) at the top or just 0.
The sum of any vector with zero vector is equal to the vector itself, i.e., if ‘a’ is any vector, then;
0+a = a
Note: There is no unit vector for zero vector and it cannot be normalised.
Co-initial and Co-terminus Vector:
Vector having same initial point are called co-initial vector , whereas vector having same terminal point are called co-terminus vector . Here a and b co-initial vector. c and d are co-terminus vector.
Equal Vector :
Two or more vectors are said to be equal vectors if they have same magnitude and direction .
- As |a| = |b| , and their direction are same regarless of initial point , we write a = b.
- Here |a| = |c| but direction are not same so. a ≠ c .
- Here direction of a and d same but |a| ≠ |d| , so a ≠ d .
Operations on Vectors
In maths, we have learned the different operations we perform on numbers. Let us learn here the vector operation such as Addition, Subtraction, Multiplication on vectors.
Addition of Vectors
The two vectors a and b can be added giving the sum to be a + b. This requires joining them head to tail.
We can translate the vector b till its tail meets the head of a. The line segment that is directed from the tail of vector a to the head of vector b is the vector “a + b”.
Characteristics of Vector Math Addition
- Commutative Law- the order of addition does not matter, i.e, a + b = b + a
- Associative law- the sum of three vectors has nothing to do with which pair of the vectors are added at the beginning.
i.e. (a + b) + c = a + (b + c)
Subtraction of Vectors
Before going to the operation it is necessary to know about the reverse vector(-a).
A reverse vector (-a) which is opposite of ‘a’ has a similar magnitude as ‘a’ but pointed in the opposite direction.
First, we find the reverse vector.
Then add them as the usual addition.
Such as if we want to find vector b – a
Then, b – a = b + (-a)
Scalar Multiplication of Vectors
Multiplication of a vector by a scalar quantity is called “Scaling.” In this type of multiplication, only the magnitude of a vector is changed not the direction.
- S(a+b) = Sa + Sb
- (S+T)a = Sa + Ta
- a.1 = a
- a.0 = 0
- a.(-1) = -a
Scalar Triple Product
The scalar tripple product , also called a box product or mixed triple product, of three vectors, say a, b and c is given by (a×b)⋅c. Since it involves dot product and evaluates single value, therefore stated as the scalar product. It is also denoted by (a b c).
(a b c) = (a×b)⋅c
The major application of the scalar triple product can be seen while determining the volume of a parallelopiped, which is equal to the absolute value of |(a×b)⋅c|, where a, b and c are the vectors denoting the sides of parallelepiped respectively. Hence,
Volume of parallelepiped = ∥a×b∥ ∥c∥ |cosϕ|=|(a×b)⋅c|
Vector Multiplication :
- Cross product
- Dot product
Cross Product of Vectors
The cross product of two vectors results in a vector quantity. It is represented by a cross sign between two vectors.
i.e., a × b
The mathematical value of a cross product-
where,
| a | is the magnitude of vector a.
| b | is the magnitude of vector b.
θ is the angle between two vectors a & b.
Dot product of Vectors
The dot product of two vectors always results in scalar quantity, i.e. it has only magnitude and no direction. It is represented by a dot (.) in between two vectors a dot b = a.b
The mathematical value of the dot product is given as
a . b = | a | | b | cos θ
Components of Vectors (Horizontal & Vertical)
There are two components of a vector in the x-y plane.
- Horizontal Component
- Vertical Component
Applications of Vectors :
Some of the important applications of vectors in real life are listed below:
- The direction in which the force is applied to move the object can be found using vectors.
- To understand how gravity uses a force of attraction on an object to work.
- The motion of a body which is confined to a plane can be obtained using vectors.
- Vectors help in defining the force applied on a body simultaneously in the three dimensions.
- Vectors are used in the field of Engineering, where the force is much stronger than the structure will sustain, else it will collapse.
- In various oscillators, vectors are used.
- Vectors also have its applications in ‘Quantum Mechanics’.
- The velocity in a pipe can be determined in terms of the vector field—for example, fluid mechanics.
- We may also observe them everywhere in the general relativity.
- Vectors are used in various wave propagations such as vibration propagation, sound propagation, AC wave propagation, and so on.
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