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What is the answer if vector A is multiplied with vector B?

In vector multiplication, there are two types of multiplication: One in which the result is a scalar and the other in which the result is a vector.

Accordingly the two types of products of vectors are:

Scalar product (also called dot product) denoted byA.B

Vector product (also called cross product) denoted by AxB

A.B  = AB cos Θ, where Θ is the angle between the directions of A and B

The physical meaning of dot product is that, “The product of the magnitude of one vector and the magnitude of the component of the other vector in the direction of the first vector.”

AxB = AB sinΘ \hat{n}

The direction of cross product is given by right hand cork screw rule

cross product

Difference between Scalar Product and Vector Product

“What is the difference between dot product and cross product?” [The question was posted by Salman)

Difference between dot product and cross product

Shashi Kant asked:

“What is difference between cross product and dot product?”

 

Answer:

If the product of two vectors is a scalar quantity it is called scalar product or dot product.

If the product of two vectors is a vector quantity it is called vector product or cross product.

A Question from Electric Flux and Gauss Theorem

Zeenath asked:

How can you prove that dS in φ =E.dS is dS Cos θ if the area is tilted at an angle ? I need the mathematical Steps.

 

Ans;

Dear Zeenath,
Electric Flux is defined as the total no of field lines passing normal to the surface. So while calculating, we need to consider the area of the surface normal to the field lines only. That is why we take the dot product of E and ds, where ds is the area vector (not just the area: Area vector is a vector whose magnitude is equal to the area and dirtected normal to the surface.)
Then by definition of dot product,
dφ =E.dS = EdS cos θ
which gives the component of E and the component of area vector in the direction of E (When area vector is in the direction of E, the actual component of area is perpendicular to E)
Hope the matter is clear.
Mathew Abraham

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