Some Terms of Vector

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

Scalar physical quantity: 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.

Vector physical quantity: 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.

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

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.

Zero or null vector: A vector having magnitude zero is called null vector. It has no specified direction and is represented by\(\vec A\).

Parallel vectors: The vectors having same direction are called parallel vectors. Here \(\vec A\) and \(\vec B\) are parallel vectors.

Equal vectors: Two vectors having same magnitude and direction are said to be equal vectors. Here \(\vec A\) and \(\vec B\) are equal vectors.

Opposite vectors: The vectors having same magnitude, but opposite direction are called opposite vectors. Here \(\vec A\) and -\(\vec A\) are opposite vectors.

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.

Coplanar vectors: The vectors lying on the same plane are called coplanar vectors. Here \(\vec A\) and \(\vec B\) are coplaner vectors

Polar vectors: A vector which produces a linear effect when acts on a body are called polar vector. Example: force, linear momentum, etc.

Axial vectors: A vector which produces turning effect when acts on a body are called axial vector. Example: angular momentum, torque, etc.

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.

Addition and subtraction of two vectors

Addition of vector

Vector quantities have both magnitude and direction so it can be added vectorically. There are two rules specially used for vector addition. They are:

1. Parallelogram law of vector addition:
   Vectors are added in accordance with the parallelogram law of addition, according to which the sum the resultant \(\vec R \) of two vetors \(\vec A\) and \(\vec B\) is the diagonal of the parallelogram of which \(\vec A\) and \vec B\) are the two adjacent sides. \(\text {i.e.}\: \vec R = \vec A + \vec B \)

   

2. The triangle law of vectors
   It states if two sides of a triangle taken in the same order represents two vectors in magnitude and direction, then the third sides represents the resultant in opposite order. Let \(\vec A\) and \(\vec B \) be the two vectors for obtaining their sum or resultant, \(\vec B\) is carried parallel to itself until the tail of \(\vec B\) coincides with the heat of \(\vec A\). Now the vetor \(\vec R\) drawn between the tail of \(\vec A\) and the heat of \(\vec B\) is the sum \(\vec A + \vec B\) as shown
   $$\text {i.e.}\: \vec R = \vec A + \vec B $$

Subtraction
Vectors are subtracted with the aid of negative vectors. The negative vector \(-\vec A \) is defined as the vector whose magnitude is the same as that of \(\vec A\) but whose direction is opposite to that of \(\vec A\).
The difference of two vectors \(\vec A\) and \(\vec B\) is defines as the sum of \(\vec A + (-\vec B)\). Thus the resultant \(\vec R = \vec A +(-\vec B)\).

Bibliography

P.B. Adhikari, Bhoj Raj Gautam, Lekha Nath Adhikari. A Textbook of Physics. kathmandu: Sukunda Pustak Bhawan, 2011.

Jha, V. K.; 'Lecture title'; Elementary Vector Analysis; St. Xavier's College, Kathmandu; 2016.