# Thread: What’s the difference between a tensor and a vector?

1. ## What’s the difference between a tensor and a vector?

I realize that all vectors are tensors, but it sure seems to me that all tensors are vectors too. But perhaps I’m missing something.

SLD  Reply With Quote

2. Learn something new every day. The dot product is a tensor, it maps product of two vectors. I always thought of it simplistically as multiple dimensions. A generalization where 2d and 3d vectors are a simplified cases.

https://en.wikipedia.org/wiki/Tensor

In mathematics, a tensor is an algebraic object related to a vector space and its dual space that can take several different forms, for example, a scalar, a tangent vector at a point, a cotangent vector (dual vector) at a point, or a multi-linear map between vector spaces. Euclidean vectors and scalars (which are often used in elementary physics and engineering applications where general relativity is irrelevant) are the simplest tensors. While tensors are defined independent of any basis, the literature on physics often refers to them by their components in a basis related to a particular coordinate system.

An elementary example of mapping, describable as a tensor, is the dot product, which maps two vectors to a scalar. A more complex example is the Cauchy stress tensor T, which takes a directional unit vector v as input and maps it to the stress vector T(v), which is the force (per unit area) exerted by material on the negative side of the plane orthogonal to v against the material on the positive side of the plane, thus expressing a relationship between these two vectors, shown in the figure (right). The cross product, where two vectors are mapped to a third one, is strictly speaking not a tensor, because it changes its sign under those transformations that change the orientation of the coordinate system. The totally anti-symmetric symbol ε i j k {\displaystyle \varepsilon _{ijk}} {\displaystyle \varepsilon _{ijk}} nevertheless allows a convenient handling of the cross product in equally oriented three dimensional coordinate systems.

Definition

Although seemingly different, the various approaches to defining tensors describe the same geometric concept using different language and at different levels of abstraction.  Reply With Quote

3. ^^^
That seems a bit overly detailed for the question.
A tensor with only magnitude (no direction) is a scalar.
A tensor with magnitude and a direction is a vector.
And then there are higher order tensors .

But a direct answer to the question is that all vectors are tensors but not all tensors are vectors. Sorta like all oaks are trees but not all trees are oaks.  Reply With Quote  Reply With Quote

5. I suggest to ignore what steve_bank wrote.
Dot product is not a a tensor, it is represented by a tensor which is redundant because pretty much everything is represented by tensors in linear algebra.

Anyway, tensor is NOT a vector. I mean not all tensors are vectors. Tensor can be viewed as direct product of n=0,1,2,3.... vectors.
In simplest non-trivial form n is 2, so it's basically direct product of two vectors (v1)x(v2).
Vector has one index, direct product of two vectors has 2 indices. In reality of course Tensor is a space of all direct products of all vectors and their sums. So not all (rank 2) tensors can be represented as a pair of vectors, most of them are sums of direct products like (V1)x(V2) + (V3)x(V4)+....

Also, Tensors are introduced after co-vectors and inner product, I suspect you skipped over that to tensors.  Reply With Quote

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