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I am new to linear algebra and have been confused by the terminology "n dimensional vector" that my online course instructor uses. He refers to vectors as an "n dimensional vector" where n is the number of elements in the vector. For example he might say that this is a 5-dimensional vector:

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However by definition a vector is 1D. I guess this is an etymology question, but why would people use such confusing terminology? If somebody said something was a "5 dimensional matrix" I assume nobody would think that means it has 5 elements, rather they would think it has 5 dimensions, so why do they talk about vectors differently?

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    This is not an etymology question, latter in your course you will learn what is a dimension. – 2017-02-20
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    In symple words a dimension is the number of elements that some vector space has in your base. The base is the set of elements of the vector space that, in some sense, generates all other elements of the space. – 2017-02-20
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    By what definition is a vector 1D? The space spanned by a single vector is certainly one-dimensional, but that’s different from the vector itself. – 2017-02-20
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    Thanks everybody for your comments, this is illuminating. As you can see I am very new to this. As for why I called a vector 1D I was just imagining that like if you have a shipping box, that's a 3D object. If you tear off one of its sides, that's a 2D object and maybe you could imagine a matrix on top of it. If you cut off a long thin part from that side, that's a 1D object and maybe you could imagine a vector on top of it. But I am only thinking colloquially and not in math terms. Looks like I have more studying to do. – 2017-02-20

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I suppose this is an old question, but as others have stated, the dimensionality of a vector refers to the space of which the vector is a member, in this case $\mathbb{R}^n$.

Though I believe what you are referring to when you say a vector is 1 dimensional is that a vector is a rank 1 tensor. Matrices are, on the other hand, rank 2 tensors, and scalar values are rank 0 tensors.

A tensor of rank $m$ can have dimensions $d_1\times d_2\times\cdots\times d_m$, where $d_i > 0$.

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    Thanks for the response, I think this clarifies things for me. – 2017-10-02
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This usually just refers to the dimension of the vector space in which the vector "lives." With that being said, this seems to be nonstandard terminology outside of $\mathbb{R}^n$.