Vector Analysis

Vector, in physics, a quantity which has both magnitude as well as direction. It is usually denoted by an arrow whose direction is the same as that of the given quantity and length is proportional to the magnitude of the quantity. Although a vector has magnitude and direction, it has no position. That is, as long as its length does not change, a vector does not change as it moves parallel to itself.

Vector Analysis

Vector analysis is a branch of mathematics which is concerned with quantities that have magnitude and direction. Some physical and geometrical quantities, known as scalar quantities, can be defined by expressing their magnitude in appropriate units of measurement. Therefore, mass is defined in grams, temperature is expressed in degrees on a scale, and time is expressed in seconds. Scalars can be represented graphically by points on a numeric scale, like a clock or thermometer. There are also some quantities, termed as vectors, which require an indication of direction and magnitude.

Vector

Vector is a mathematical structure. It has many uses in physics as well as in geometry. We know that the position of points on the coordinate plane can be expressed by the ordered pair as (x, y). Using vectors is very useful in simplifying three-dimensional geometry.

When we measure a physical quantity, the measured value we get is always a number. This number only makes sense if the corresponding unit is specified. The result of a measurement therefore has a numerical value and a unit of measurement.

Vector Quantity

Vector quantity is a physical quantity which has magnitude with certain unit and also a specific direction. Therefore, specifying the direction of action with its magnitude is necessary when we define or state a vector quantity. Displacement, weight, force, velocity, and more are vector quantities which have both direction as well as magnitude.

Examples of Vector Quantity

There are many examples of vector quantities in which some important quantities are given here. 

Velocity

Velocity has magnitude and also direction. Velocity is the first derivative of the displacement with respect to time in calculus. Velocity is also defined as the rate of change of position of an object w.r.t time. 

Unit of velocity is given as a meter per second which is expressed as m/s.

Velocity=displacement/time

velocity=final position-initial position/time

Displacement

The displacement is the change in position of a body with respect to the reference frame. For example, when a bike is moving from a house to a grocery store, then its displacement is the relative distance from grocery store to the house.

Formula for displacement is given as

∆x =x_f–x_i

Here,

∆x = displacement 

x_f= final position

x_i= initial position

Acceleration

Acceleration is the rate of change of velocity with respect to the time. 

Formula for acceleration is given as

a=dv/dt

Vector Addition

The process of addition of two or more vectors is considered as vector addition. On the basis of direction of vector, vector addition is divided into two types which are as follows

1. Parallelogram Law of Vector Addition
2. Triangular Law of Vector Addition

Parallelogram Law of Vector Addition

According to Parallelogram Law of Vector Addition, when two vectors are adjacent sides of a parallelogram then the resultant of two vectors is the vector which passes diagonally through the point of contact of two vectors.

Triangular Law of Vector Addition

When two vectors are arranged in such a way that the head of one vector is connected with the tail of another vector then the triangular law of vector addition is applicable.

According to the Triangular Law of Vector Addition, if two vectors are expressed by two sides of a triangle in magnitude and direction which are considered in the same order, then the third side of that triangle represents the magnitude and direction resultant vectors.

Uses of Vector

There are many uses of vectors which are given below.

1. Vectors are applicable in quantum mechanics.
2. Vector is also used in oscillators.
3. Vectors are used in the field of engineering.
4. Vector is used in game and also in designing roller coasters.

Polar Vectors: 

Polar vector has an initial point, like displacement and force.
Axial Vectors: 

Axial vectors define rotational effect and work along the rotational axis on the basis of right-hand screw rule, like angular velocity, torque, and also angular momentum.

Conclusion

Vector analysis is a branch of mathematics which is concerned with quantities that have magnitude and direction.

Vector has many uses in physics as well as in geometry

Vector quantity is a physical quantity which has magnitude with certain unit and also a specific direction.

Displacement, weight, force, velocity, and more are vector quantities which have both direction as well as magnitude.

Velocity is also defined as the rate of change of position of an object w.r.t time.

Velocity=displacement/time

velocity=final position-initial position/time

The displacement is the change in position of a body with respect to the reference frame.

Vector addition is divided into two laws which are as follows

1. Parallelogram Law of Vector Addition
2. Triangular Law of Vector Addition

Vectors are applicable in quantum mechanics.

Vector is also used in oscillators.