Vector: An Introduction to the Basics (Oversimplified)

Vector: The physical quantities which require both magnitude and direction for complete description are called vectors. eg:- moment, impulse, angular displacement, angular velocity, angular acceleration, torque, magnetic moment, gradient of any quantity, length element, area element, etc.
Scalar: The physical quantities which require only magnitude for complete description are called scalar quantity. eg:- length, distance, area, flux of vector quantity, intensity of radiation, electrical charge, electrical current, electrical potential, etc.
Tensors: Physical quantities having no fixed direction but different magnitude at different points are called tensors. eg:- moment of inertia, electrical conductivity, dielectric constant, refractive index, etc.
Condition for a quantity to be a vector:
Necessary Condition
The quantity should have a particular direction.
Sufficient Condition
It should follow triangle law of vector addition.
Types of vector:
Composition of vectors: The process of combining two or more vectors into a single vector which is the resultant of them is called composition of vectors.
Resultant of vectors or resultant vectors: A single vector that produces same effect as is produced by individual vectors together is called resultant of vectors or resultant vector.
Angle between two vectors: The angle between two vectors is measured by putting their tails together. If 2 vectors are not acting on a point, then angle between them can be obtained by displacing one of the vector parallel to itself so that the tails of 2 vector meets at a point.
Here, the angle between vectors A and B is the the angle β.
Vector Representation:
One way to represent a vector is by means of an arrow. The direction of the arrow is the direction of the vector and the length of the arrow, drawn to a scale represents its magnitude.
Addition of Vectors:
Triangle Law of Vector Addition
If 2 vectors, P and Q are represented in magnitude and direction by 2 adjacent sides of a triangle taken in same order then their resultant vector, r(= p+q) is represented in both magnitude and direction by the 3rd side of the triangle in opposite order.
Let OA and AB are two p and q vectors respectively.The magnitude and direction of the resultant vector (r) of vectors p and q is and with respect to vector p respectively.
Parallelogram Law of Vector Addition
If 2 vectors acting simultaneously at a point can be represented by 2 adjacent sides of a parallelogram drawn from a point then their resultant is represented in both magnitude and direction by the diagonal of parallelogram drawn from that point.
Let 2 vectors OA(p) and OB(q) initializing from the sameThe magnitude and direction of the resultant vector (r) of vectors p and q is and
with respect to vector p respectively.
Parallelogram Law of Vector Addition
If a number of represented in both magnitude and direction by (n - 1) sides of n sided polygon taken in same order, then the nth side (closing side) of this polygon taken in opposite order represents the resultant of them in both magnitude and direction.
Law of Algebra for the Vector Sum:
Commutative Law:
i.e.,
Associative Law:
i.e.,
Subtraction of Vectors
The process of adding of a vector with negative of another vector is called vector subtraction. In order to find out the magnitude and direction of vector a-b, we reverse the direction of vector b thus producing –b vector which is added to vector a and the resultant vector is the subtraction of vector b from a.
Multiplication of Vectors
Multiplication of a Vector by a Scalar
If vector A is multiplied by a scalar quantity m, the product mA is a vector that has same direction as A and magnitude m×|A|.
The direction may change as follows:
Laws of Algebra for Scalar Multiple
Multiplication of a Vector by a Vector
There is 2 kind of multiplication of a vector by a vector:
Multiplication of a vector by a second vector by a second vector so as to produce a scalar is called scalar or dot product of given 2 vectors.
Consider vectors A and B with an angle Ɵ between them as shown in figure. Then, the scalar product of them is given by
Since, A, B, cosƟ are scalars, product of vectors A and B is also scalar. Being scalar it has no direction.
It is also defined as the product of magnitude of one vector and scalar components of second vector along the direction of first vector.
Properties of Dot Product:
Laws of Algebra for Dot Product:
Multiplication of a vector by another vector so as to produce another vector is called cross product of 2 vectors.
It is denoted by cross(×) between 2 vectors. It has both magnitude and direction, so it is also called vector product.
Consider vectors A and B with smaller angle Ɵ between them. Then the cross product of vectors A and B is given by vector C(= A×B). It is a vector because C is itself a vector. So, it has both magnitude and direction.
Direction of vector A×B
Its direction is along the line perpendicular to the plane containing vector A and vector B and is given by right hand thumb rule.
Properties of Cross Product
Laws of Algebra for Cross Product:
The division of any 2 vectors is not possible.
Right Hand's Thumb Rule:
Curl the fingers of the right hand so that it would rotate first vector into the second vector through the smaller angle between them then the extended thumb gives the direction of vector C(= A×B).
Where n is unit normal vector along the direction of vector A×B which is perpendicular to both the vectors and to the plain containing two vectors.
Resolution of Vectors: The process of splitting a vector into 2 or more vectors in different directions on a plane such that their sum gives back the original vector is called the resolution of vectors.
The splitted vectors are called components of the vector.