Advertisement

## dot product of two vectors

When two vectors are multiplied with each other and answer is a scalar quantity then such a product is called scalar product or dot product of vectors.

A dot (.) is placed between vectors which are multiplied with each other that’s why it is also called “dot product”.

**Scalar = vector .vector**

## Vector dot product example

- The product of force
**F**and displacement**S**is work “W”.

i.e W =**F** **.** **S**

- The product of force F and velocity V is power “P”.

i.e P =**F** **.** **V**

- The product of electric intensity E and area vector A is electric flux Φ.

i.e Φ = **E . A**

## Dot product formula

The product of magnitudes of vectors and the cosine of angle between them.Consider two vectors A and B making an angle θ with each other.

** A . B** = AB Cos θ

Where “B Cos θ ” is the component of **B** along vector **A **and 0 ≤ θ ≤ π.

## Dot product properties

- If vector
**A**is parallel to**B**then their scalar product is maximum.

i.e **A . B** = AB Cos 0º=AB (1) =AB

- Scalar product of same vectors is equal to square of their magnitude.

** A . A** = AA Cos 0º=A² (1)=A²

- If two vectors are opposite to each other than their scalar product will be negative.

i.e **A . B** = AB Cos 180º=AB (-1) = -AB

- If vector A is perpendicular to B then their scalar product is minimum.

i.e **A . B** = AB Cos 90º=AB (0) = 0

- For unit vectors i ,j and k ,the dot product of same unit vectors is 1 and for different unit vectors is zero.

**i.e i. i** = **j . j** = **k . k** = 1

and

** i. j** = **j . k** = **k . i** = 0

## Vector cross product

“When two vectors are multiplied with each other and the answer is also a vector quantity then such a product is called vector cross product or vector product.”

A cross (×) is placed between the vectors which are multiplied with each other that’s why it is also known as “cross product”.i.e

Vector = Vector × Vector

## Vector cross product example

- The product of position vector “
**r**” and force “**F**” is Torque which is represented as “**τ**“.

i.e ** τ** = **r** × **F**

- The product of angular velocity
**ω**and radius vector “**r**” is tangential velocity.

i.e **V** t = **ω × r**

## Cross product formula

The cross product is defined by the relation

**C** = **A** **× B = AB **Sinθ **u**

Where **u **is a unit vector perpendicular to both A and B.

**Related topics in our site:**

- Difference between dot product and cross product
- Types of vectors
- Difference between vector and scalar quantities