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dot product and cross product of vectors with examples and properties

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 What is 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 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 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.

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

  • 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

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.

 

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