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What is a tensor?

Stress is tensor,then,what is meant by tensor? (Rupasri asked)

Answer:

Tensor is a mathematical object analogous to but more general than a vector, represented by an array of components that are functions of the coordinates of a space.

Tensors are used to represent correspondences between sets of geometrical vectors.

Refer to the following links for a more detailed analysis.

http://www.grc.nasa.gov/WWW/k-12/Numbers/Math/documents/Tensors_TM2002211716.pdf

http://www.youtube.com/watch?v=HDCafA-Xjis

 
Physically, vectors are used to represent locations, velocities, accelerations, flux densities, field
quantities, etc.

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)

Examples of scalar product and vector product

Kashan asks:

“will you give me min 10 examples 0f scalar product and vector product known at xii level”

I am just strating the list; members please contribute more as comments

Scalar Product

  1. work F.s
  2. Power F.v
  3. Potential energy of a dipole P.E

 

Vector Product

  1. Torque on a dipole = PxE
  2. Lorentz magnetic force F=qvxB
  3. Torque =rxF

 

 

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

Rain and Man problems – How to solve?

Shashank Asked:

“How to solve rain and man problems from motion in one dimension?”

Answer:

The problem is solved using principles of vector addition.

There is a simple logic to remember

A/C = A/B x B/C

If we represent diagrammatically (in the form of vectors) the velocity of rain wrt ground, velocity of man wrt ground and velocity of rain wrt man, then we can write

Vr/m = V r/g + Vg/m = V r/g – Vm/g = V r/g + (- Vm/g)

Where (- Vm/g) is the negative vector of Vg/m

I will be adding illustrations soon.

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