system
Given the following post, respond as if you're commenting on Reddit:
human
What do the dot product/the tensor product of two vectors represent geometrically?

So the cross product isn't too hard to understand conceptually, since there's an actual geometric representation for what it is (if a x b = c, then c is perpendicular to a and b and has a magnitude equal to the area of the parallelogram made by a and b (E: that goes in the direction according to the right-hand rule)). But what about the dot/tensor product? I know how to do them, but what exactly do they represent geometrically?
gpt
The dot product of **u** and **v** is |**u**||**v**|cos(θ), where θ is the angle between **u** and **v**.

The tensor product of **u** and **v** is a bit more complicated. This is also called the outer product of **u** and **v**. If we represent **u** as a column vector, then the outer product **u** ⊗ **v** is the matrix **u****v**^(T). Note that (**u** ⊗ **v**)**x** = (**v · x**)**u**, which is just a multiple of the fixed vector **u**. Hence **u** ⊗ **v** is a rank-1 linear mapping from **R**^(n) to **R**^(n), which means the image is one-dimensional. Its image is just the span of the vector **u**, i.e., the line through the origin containing the vector **u**. So **u** ⊗ **v** is a projection (although it is not necessarily an orthogonal projection). The transformation **u** ⊗ **v** generally projects a vector **x** onto the line spanned by **u** and then multiplies it by some scalar (that depends on **x**).

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Also, for the cross product, your description is not enough to uniquely define the cross product since there are two possible vectors with the two properties you have described. You must also declare that the direction of **a** x **b** is determined by the right-hand rule. This means that if **a** and **b** are linearly independent, then the ordered basis (**a**, **b**, **a** x **b**) is right-handed, i.e., the determinant of the matrix whose columns are **a**, **b**, and **a** x **b** is positive.