Introduction to Vector and Scaler

Physical quantity

A measurable quantity that defines the laws of physics is known as physical quantity. The physical quantities are divided into two types:

1. Scalar physical quantity: Certain physical quantities are completely described by numerical value alone (with units specified) and are added according to ordinay rules of algebra. As an example the mass of a system is described by saying that it is 3 kg. If two bodies one having a mass of 3 kg and another having a mass 4 kg are added together to make a composite system, the total mass of system becomes 3 kg+4 kg = 7 kg. Such quantities are calledscalars.A physical quantity which has a magnitude but no direction is called scalar quantity. Example: length, mass, time, area, temperature, electric current, pressure, work, energy, electric flux, magnetic flux, etc. The scalar quantities can be added or subtracted by using a simple algebraic method. [If the change in scalar quantity is very small i.e. →0, then this change can be regarded as a vector quantity].
2. Vector physical quantity: The complete description of certain physical quantities a numerical value (with units specified) as well as a direction in space . Velocity of a particle is an example of this kind.A physical quantity which has both magnitude and direction is called vector quantity. Example: displacement, velocity, acceleration, force, weight, momentum, impulse, magnetic field, electric field, gravitational field, etc. The vector quantities cannot be added or subtracted by simple algebraic method but can be added or subtracted by using geometric method such as triangle law of vector, parallelogram law of vector, polygon law of vector, etc.

Graphical representation of a vector

A vector quantity can be represented graphically by the means of a straight line with an arrow as shown in the figure. The length of the straight line gives the magnitude of the vector and the arrow gives its direction. The starting point like 'O' is called tail and the ending point like 'A' is called head.

Notation of a vector

A vector quantity can be written as alphabet (small or capital) with arrow-head like

Example:\(\overrightarrow {OA}, \overrightarrow {OB},\overrightarrow {OC}\)

\(\vec a, \vec b,\vec c\)

\(\vec A, \vec B,\vec C\)

Difference between Scalar and Vector Quantities

Types of vector

1. Unit vector: A vector having magnitude 1 unit is known as a unit vector. The unit vector can be written as an alphabet with hat or cap. Example:$$\widehat A = \frac{\vec A}{\lvert A \rvert}$$The direction of unit vector is along its vector.
2. Zero or null vector: A vector having magnitude zero is called null vector. The direction of a zero vector is indeterminate and is represented by\(\vec A\).
3. Parallel vectors: The vectors having same direction are called parallel vectors. Here \(\vec A\) and \(\vec B\) are parallel vectors.
4. Equal vectors: Two vectors having same magnitude and direction are said to be equal vectors. Here \(\vec A\) and \(\vec B\) are equal vectors.
5. Opposite vectors: The vectors having same magnitude, but opposite direction are called opposite vectors. Here \(\vec A\) and -\(\vec A\) are opposite vectors.
6. Collinear vectors: The vectors passing through the same straight line are called collinear vectors.Here \(\vec A\) and \(\vec B\) , \(\vec P\) and \(\vec Q\) are collinear vectors.
7. Coplanar vectors: The vectors lying on the same plane are called coplanar vectors. Here \(\vec A\) and \(\vec B\) are coplaner vectors
8. Polar vectors: A vector which produces a linear effect when acts on a body are called polar vector. Example: force, linear momentum, etc.
9. Axial vectors: A vector which produces turning effect when acts on a body are called axial vector. Example: angular momentum, torque, etc.
10. Proper vectors: The vectors other than null vectors are called proper vectors.

The necessary condition for a physical quantity to be a vector: Must have magnitude and direction.

Sufficient condition: To obey the vector law of addition.