Inner product of rank 2 tensor and a vector

In summary, the conversation discusses the inner product of a dyad and a vector, which can result in a vector at an angle to the initial one. The cross product can also be used, but only if a right angle is desired. The conversation also mentions encountering a situation where the inner product of a vector and a dyad results in a vector of different magnitude but same direction. The question is raised about the conditions and properties of such an interaction, using a 2D version of a matrix and a vector as an example.
  • #1
abluphoton
20
0
I been reading some material that lead me to understand that it takes an inner product of a dyad and a vector to obtain another vector at an angle to the initial one... cross product among two vectors would be an option only if we are willing to settle to a right angle.
After few days i countered a situation where i see an inner product of a vector and a dyad resulting in a vector of different magnitude but same direction as the earlier one. i mean to say
[ A*v = η.v ]
A= a random dyad
v= a vector
η= a scalar.

now the question is, what is the condition for such an interaction ?? what should be the property of such a tensor ?!
 
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  • #2
look at a 2D version of a matrix and a vector say M x V = n x V

M x V = results in:

n*vx=m11*vx + m12*vy

and

n*vy = m21*vx + m22*vy

so what could you set the m values to make the two equations true?
 

1. What is the definition of an inner product of a rank 2 tensor and a vector?

The inner product of a rank 2 tensor and a vector is a mathematical operation that takes two mathematical objects, a tensor of rank 2 and a vector, and produces a scalar value. It is a generalization of the dot product between two vectors.

2. How is the inner product of a rank 2 tensor and a vector calculated?

The inner product of a rank 2 tensor A and a vector B is calculated by multiplying the components of A with the components of B and then summing the products. This can also be represented as AijBi where i and j represent the indices of the tensor and vector, respectively.

3. What is the significance of the inner product of a rank 2 tensor and a vector?

The inner product of a rank 2 tensor and a vector is useful in many areas of mathematics and physics. It allows us to define a metric for measuring distances and angles in a vector space, and it is also used in calculations involving tensors and vectors, such as in mechanics and electromagnetism.

4. Can the inner product of a rank 2 tensor and a vector be negative?

Yes, the inner product of a rank 2 tensor and a vector can be negative. This can occur when the vectors are not aligned in the same direction, resulting in a negative dot product. However, the inner product is always a real number.

5. How does the inner product of a rank 2 tensor and a vector differ from the dot product of two vectors?

The inner product of a rank 2 tensor and a vector is a generalization of the dot product between two vectors. While the dot product only operates on two vectors, the inner product can operate on a tensor of any rank and a vector. Additionally, the inner product takes into account the direction and magnitude of the vectors, while the dot product only considers their magnitudes.

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