IEMDaily - Video Lecture Notes

Definition

Scalar Quantities

A scalar quantity is a quantity which is defined by only magnitude. Some examples of scalar quantities are Mass, Charge, Pressure, etc.

Definition

Vector quantities

Vector quantities are those which have both magnitude and direction and obey vector laws of addition. Some examples of vectors are displacement, velocity, force, etc.
A quantity is called a vector only if it follows all the above three conditions. For example, current is not a vector despite having both magnitude and direction because it does not follow vector laws of addition.

Definition

Scalar

Scalar Quantities:
1. A physical quantity which is having magnitude only but not direction those are scalar quantities.
2. These are one-dimensional quantities.
3. It follows ordinary rules of algebra.
4. The scalar can be divided by any other scalar quantity.
5. It changes due to change in their magnitude only.
6. e.g. mass, work etc

Definition

vector

Vector Quantity:
1. A vector quantity is one, that has both magnitude and direction.
2. Are multi-dimensional quantities.
3. It changes with the change in their direction or magnitude or both.
4. Follow rules of vector algebra.
5. Two vectors can never divide.

Diagram

Direction of vector

Vector

\vec{a}

shown in diagram can be represented as

\vec{a} = a_{x} \hat{i} + a_{y} \hat{j} + a_{z} \hat{k}

Magnitude of vector a
a =

\sqrt{a_{x}^{2} + a_{y}^{2} + a_{z}^{2}}

a_{x}

- component of vector along x-axis

a_{y}

- component of vector along y-axis

a_{z}

- component of vector along z- axis

\hat{i}, \hat{j} a n d \hat{k}

- unit vectors along x,y and z direction.

Result

Scalars and Vectors

Following are some differences listed between scalars and vectors.

S.No.	Scalars	Vectors
1.	Have only magnitude	Have both magnitude and direction
2.	Algebra: Same as real numbers	Algebra: Follow vector laws of addition
3.	Examples: Mass, charge, etc	Examples: velocity, force, electric field. etc.

Definition

Free and fixed vectors

Free vector:
A free vector is a vector whose action is not confined to or associated with a unique line in space. Examples: velocity vector, electric field vector, etc.
Sliding vector:
A vector that can be applied at any point on a body as long as it is along its original line of action and doesn't change its effect in the body as a whole. Example: force vector, etc.
Fixed vector:
Fixed vector is that vector whose initial point or tail is fixed. Example: position vector, etc.

Definition

Difference between scalar and vector

	SCALAR QUANTITY	VECTOR QUANTITY
Meaning	Any physical quantity that does not include direction is known as scalar quantity.	Vector quantity is one, that has both magnitude and direction.
Quantities	One-dimensional quantities	Multi-dimensional quantities
Change	It changes with the change in their magnitude.	It changes with the change in their direction or magnitude or both.
Operations	Follow ordinary rules of algebra.	Follow rules of vector algebra.
Comparison of two quantities	Simple	Complex
Division	Scalar can divide another scalar.	Two vectors can never divide.

Definition

Axial Vector

Axial vector is a vector which does not change its sign on changing the coordinate system to a new system by a reflection in the origin.
An example of an axial vector is the vector product of two polar vectors, such as A = x $\times$ m, where A is the angular momentum of a particle, x is its position vector, and m is its momentum vector.

Definition

Equality of vectors

Two vectors are said to be equal if
(i) they have the same magnitude and
(ii) are in the same direction.
If we shift B parallel to A then it will completely superimpose A i.e it has same length and are in the same direction as A, so