TITLE: What really is the difference between $\mathbf{v}=(2,3) $ and $\mathbf{u}=(2,3,0)$?
QUESTION [5 upvotes]: I used to believe that there is no difference between $\mathbf{v}=(2,3)$ which is a vector lying in $xy$ plane and $\mathbf{u}=(2,3,0)$ which is a three dimensional vector but still lying in $xy$ plane. So, for me both the vectors $\mathbf{v}$ and  $\mathbf{u}$ were same. The analogy I used was that both the tail and head coincide for $\mathbf{v}$ and $\mathbf{u}$.
But on the second page of the textbook I am using as reference (Gilbert Strang-Introduction to linear algebra), I found something different . The author says, 
The vector $(x,y)$ in a plane is different from vector $(x,y,0)$ in $3-$space.
That's it. Since this statement is not well explained there so I am here. Hoping for help. 
Since I am beginning the Linear algebra course so Please show some tolerance. Thanks.

REPLY [3 votes]: The vector $(2, 3)$ is an element of the two-dimensional vector space $\mathbb R^2$. The vector $(2, 3, 0)$ is an element of the three-dimensional vector space $\mathbb R^3$. What is the difference between the two?
There are infinitely many ways of embedding $\mathbb R^2$ in $\mathbb R^3$. By "embedding," we mean an injective isomorphism of vector spaces. The image of an embedding $\mathbb R^2 \to \mathbb R^3$ is just a plane in $\mathbb R^3$ which contains the origin $(0, 0, 0)$, so the choice of an embedding is (sort of) equivalent to the choice of a plane through the origin. There are infinitely many such planes, and each plane can be thought of as a "copy" of $\mathbb R^2$ sitting inside of $\mathbb R^3$.
When you think of $(2, 3)$ as the same as $(2, 3, 0)$, what you are really doing is thinking of $\mathbb R^2$ as the $(x, y)$-plane in $\mathbb R^3$, as you said. In this case you have chosen to embed $\mathbb R^2$ in $\mathbb R^3$ via the map $(x, y) \mapsto (x, y, 0)$. But there are infinitely many other embeddings you could have chosen.
I think the point the author of your textbook is trying to make is this: elements of two-dimensional vector spaces are not elements of three-dimensional vector spaces. If you want to consider elements of $\mathbb R^2$ as elements of $\mathbb R^3$, you have to choose an embedding.