What really is the difference between $\mathbf{v}=(2,3) $ and $\mathbf{u}=(2,3,0)$? 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.
 A: They're different because they're members of different vector spaces. After all, the vector $(2,3)$ is defined with respect to the standard basis $e_1, e_2$; what's to say that the vector $(2,3,0)$ has been defined wrt a basis whose first two vectors are $e_1, e_2$ and whose third vector is orthogonal to those two?
If you wanted to abuse notation, you could write that $(2,3) = (2,3,0)$. But this is so non-standard that it would confuse everyone who read it. There's no good reason, for instance, that you should miss out the third component rather than the second: why don't you have $(2,3) = (2,0,3) = (2,0)$?
Vector spaces often don't canonically embed into other vector spaces, so there's often no canonical way of identifying a vector in one space with a vector in another.
A: 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.
A: They are different because for example we can't add or subtract them. Try ;)
