Mechanics

# Dot product and Cross product of vectors

Dot product and cross product are two types of vector product. The basic difference between dot product and cross product is that dot product always gives scalar quantity while cross product always vectors quantity. The dot product is always used to calculate the angle between two vectors.

## What is the dot product of two vectors?

When two vectors are multiplied with each other and the answer is a scalar quantity then such a product is called the 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 examples

• 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

### The dot product formula

The product of magnitudes of vectors and the cosine of an 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 with vector A and 0 ≤ θ ≤ π.

### Scalar 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

• The scalar product of the same vectors is equal to the square of their magnitude.

A . A = AA Cos 0º=A² (1)=A²

• If two vectors are opposite to each other then 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 the same unit vectors are 1 and for different units, vectors are zero.

i.e                                    i. i = j . j = k . k = 1
and
i. j = j . k = k . i = 0

## What is Vector cross product of two vectors?

“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

### Examples of  Vector cross product

• 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 is a unit vector perpendicular to both A and B.

### Dot product and cross product of two vectors (video)

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