"a three-dimensional vector can point in any physical space".
Or not. Homogenous 2D, then all '3d' vectors are in a plane. Talking about dimensions is always a pain.
Technically, a vector can have only one dimension. Such a vector is called a scalar.
Vectors always have a direction and magnitude. (vector length is just one posible interpretation of its magnitude BTW) Scalars always have a magniude, but never a direction. (I should probably say orientation instead of direction, but even I'm not that anal) A one dimensional vector acts as a pseudoscalar in a one dimensional space, which if diffrent from scalar...not that I'd talk about any of this in an intro presentation. (An example of a pseudoscalar is the imaginary part of a complex number...which is a bivector..fun, huh?)
I don't like that when talking about positions that they are displayed as points...it's misleading. I'd prefer that the vectors from the orgin were shown. I've no clue why the Hadamaard product is even shown, as most people will never use it. And why not have the inner & outter products (dot & cross)? Without them your pretty much talking about ordered sets.