What is scalar and vector quantity?

The physical quantity which have only magnitude but no direction, are called scalar quantity.
Mass, length, time, speed, volume, density, pressure, temperature, work, energy, power, electric current, electric charge, electric potential, electric flux etc are the examples of scalar quantity.

Scalar or Dot Product

Dot product of two vectors A and B is represented by,
A .B = AB cosθ


Where θ is angle between two vectors A and B.

•   If two vectors A and B are parallel, then θ = 0^०


∴A.B = AB
For unit vectors,

î . î = ĵ . ĵ = k.k = 1


•   If two vectors A and B are mutually perpendicular, then θ = 90^०

∴A.B = 0

For unit vectors,

î . ĵ = ĵ . k = k.i = 0


•   If two vectors A and B are anti parallel, then θ = 0^०


∴A.B = – AB

•   Properties of dot product


1.Dot product of two vectors is commutative.

A . B = B . A

2.Dot product is distributive.

A . ( B + C ) = A . B + A . C

•   Dot product of two vectors A and B in component form
A . B = A_xB_x + A_yB_y + A_zB_z

Cross Product of two vector

Cross product of two \( \vec{A}\) and \( \vec{B}\) is represented by,

$$ \vec{A} \times \vec{B} = A B \sin \theta \hat{n}$$

Where \( \hat{n}\) is the unit vector along the resultant vector.

If two vectors \( \vec{A}\) and \( \vec{B}\) are parallel,

Then $$ \theta = 0^o or 180^o$$
So $$ \vec{A} \times \vec{B} = 0$$

For unit vectors

$$ \hat{i} \times \hat{i} = \hat{j} \times \hat{j} = \hat{k} \times \hat{k} = 0$$

If two vectors \( \vec{A}\) and \( \vec{B}\) are perpendicular,

Then $$ \theta = 90^0$$
So $$ \vec{A} \times \vec{B} = AB \hat{n}$$

For unit vectors
\( \hat{i} \times \hat{j} = \hat{k}\) , \( \hat{j} \times \hat{k} = \hat{i}\) , \( \hat{k} \times \hat{i} = \hat{j}\)

Properties of cross product

1. Cross product of two vectors in not commutative.
   $$ \vec{A} \times \vec{B} = – \vec{B} \times \vec{A}$$
2. Cross product is distributive.
   $$ \vec{A} \times ( \vec{B} + \vec{C} ) = \vec{A} \times \vec{B} + \vec{A} \times \vec{C}$$

Cross product of two vectors \( \vec{A}\) and \( \vec{B}\) in component form

$$ \vec{A} \times \vec{B} = ( A_y B_z – A_z B_y ) \hat{i} +
( A_z B_x – A_x B_z ) \hat{j} + ( A_x B_y – A_y B_x ) \hat{k}$$