# Scalar and Vector

Physical quantities can be classified into two categories, which are scalars and vectors. Quantities like mass or density can be described by their numerical values and appropriate units only. These quantities are called “scalars”. However, quantities like velocity or force require the specifications of a numerical value and a direction. For example, specifying the value of velocity is not enough to understand an object’s motion. It is necessary to mention the direction of its motion. Such quantities are referred to as vectors. The physical interpretations, algebra, and calculus are very different for the two types of quantities.

### Scalar Quantity Definition

A scalar quantity only has a magnitude and it can be represented by a number only. A scalar does not have any direction. The addition of scalars follows the generic rules of the addition of numbers.

### Vector Quantity

A physical quantity, having both magnitude and direction, is referred to as a vector. The addition of two vectors does not follow ordinary algebra. A vector quantity is represented with an arrow over a letter or a boldface letter. Geometrically, it is represented by a line segment, having an arrow at one end. The arrow describes the direction and the length of the segment gives the magnitude.

### Examples of Scalar and Vector Quantities

Some common examples of scalar quantities are mass, time, speed, volume, temperature, density, and many more.

Displacement, velocity, acceleration, momentum, force, weight, etc. quantities are represented by vectors.

### Laws of Addition of Vectors

Vector addition can be defined using any of the following laws,

Triangle Law: If two vectors are denoted by the sides of a triangle in the same order, the resultant vector is given by the third side of the triangle, taken in the opposite order.

Parallelogram Law: If two vectors are denoted by two adjacent sides of a parallelogram, the resultant vector is given by the diagonal that passes through the point of intersection of those sides.

The resultant (addition) of two vectors a and b with magnitudes a and b is given by,

c = a + b

The resultant vector c has magnitude,

c = $\sqrt{a^{2}+b^{2}\;2ab\; cos \alpha}$

It makes an angle with the vector a such that,

tan$\theta$ = $\frac{bsin\alpha}{a+b\;cos\alpha}$

Vector subtraction can be expressed as addition of the inverted vector to be subtracted.

### Force is Scalar or Vector?

Force is a quantity that can change the state of motion of an object. It has both magnitude and direction. The SI unit of force is Newton (N).

### Mass is Scalar or Vector?

Mass is a scalar quantity. It is a measure of the inertia of an object. Mass can be represented by a number only. The SI unit of mass is kg.

### Weight is Scalar or Vector?

Weight is a vector quantity. It is given by the amount of force exerted on an object, due to a gravitational force. The weight of an object on the Earth has a direction towards the center of the Earth. SI unit of weight is Newton (N).

### Displacement is Scalar or Vector?

Displacement of an object is given by the straight distance traversed by the object at any given time interval. It is a vector quantity and it points from initial to final position of the body within that interval of time. The SI unit of displacement is meter (m).

### Speed is Scalar or Vector?

Speed of an object is a scalar but velocity is a vector. Velocity has a direction as that of displacement. Velocity points in the direction of motion. The SI units of both speed and velocity are m/s.

### Acceleration is Scalar or Vector?

Acceleration of an object is caused due to the change of velocity of the object. It is a vector quantity having unit m/s2.

### Area is Vector or Scalar?

Area is a vector quantity. It has a magnitude equal to the amount of space inside any boundary. The normal direction to that space is associated with the area. The SI unit is m2.

### Pressure is Scalar or Vector?

Pressure is the amount of normal force per unit area. It is a scalar quantity however force is a vector. Pascal (Pa) or N/m2 is the SI unit of pressure.

### Work is Scalar or Vector?

Work is the energy associated with a force. If a force acts on a body and the body undergoes a displacement, the amount of work done is the product of force and displacement parallel to the force. Work has the dimensions of energy, which is also a scalar. The SI unit of work is Joule (J).

### Did You Know?

• When an object moves along a path joining two points, the distance is measured along the trajectory whereas displacement is the shortest path joining the two points. Consequently, distance varies if the object follows different trajectories between the initial and final positions. However, the displacement between two fixed positions is independent of the path followed by the object. Distance is a scalar, however, displacement is a vector.

• Speed and velocity are closely related but different concepts. For example, the speed of an object remains constant throughout a uniform circular motion but the velocity is different at every point since the direction of velocity changes.

• The weight of a body depends on its mass. Although the mass of an object remains the same, its weight can vary due to variation in the gravitational field.