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_{x}B_{x} + A_{y}B_{y} + A_{z}B_{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

- Cross product of two vectors in not commutative.

$$ \vec{A} \times \vec{B} = - \vec{B} \times \vec{A}$$ - 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}$$