Well, ok, but do mention that the dot-product can be seen as a tensor itself; ie a (0,2)-tensor. You give in two vectors and this tensor gives back a scalar.Antiphon said:If you want to know how much of one vector points along the direction of another
vector you use a dot product.
I am not sure this is quite accurate. i think you should replace the dot product by tensor product. I mean, the dot product is defined as a (0,2)-tensor so it must always yield a scalar. You are referring to a matrix product (which is also a tensor) or more generally a tensor-product.But a tensor is something which can result in another vector when you take the dot
product of the tensor and the vector.
The vector you get out can point in a different
Well this is ofcourse true but this is certainly not a very general definition of a tensor. I mean, you can change the direction of a vector by simply multiplying it by a scalar : -1 for example. Indeed a scalar is also a tensor in itself, but the point is that i can apply your way of reasoning without using the word tensor ONCE. Hence this is not complete. Besides you are only giving 50% of the definition at best because you are totally omitting the required transformation properties. In your case Christoffel symbols would also be tensors but THEY ARE NOT. I explain this i my above journal entryTo change it's direction requires multiplying it with
What exactly are you talking about here ? A different direction to what ?