# What does this vector notation really mean?

With regard to vectors, how is this (form 1): $$\begin{bmatrix}1\\2\\-1\end{bmatrix}$$ Different to this (form 2): $$\begin{bmatrix}1\ 2\ -1 \end{bmatrix}$$
I would think that the first set consists of magnitudes of the same variable (where simply the sum gives the magnitude), while the second refers to the coefficients of three different variables.

In my book they take the magnitude of form 1 by $$\sqrt{1^2+2^2+(-1)^2}$$ That would indicate a set of vectors in different directions. So my question is; which of the two (rows or columns) give an indication of the variable or the vector direction / dimension?

• What is the difference between these: $\mathcal{7}$ $\mathbf{7}$ $7$ 7 – vadim123 May 15 '15 at 14:13
• @vadim123 C'mon, this is a valid question. A similar one might be Is “a+0i” in every way equal to just “a”?. – user1729 May 15 '15 at 14:17
• @user1729, my comment was not meant to denigrate the question, but to answer it. The difference is one of notation only. The link you provide is a substantially different question. $7$ is an element of both $\mathbb{Q}$ and $\mathbb{C}$, but $7+0i$ is an element of $\mathbb{C}$ only. – vadim123 May 15 '15 at 14:24
• You might also be interested in math.stackexchange.com/q/1198729/139123 and its answer, although I think you have some questions about the meanings of the individual numbers within a column vector that are not really considered there. – David K May 16 '15 at 4:04

That being said, triples of real numbers from a vector space (with suitable "obvious" operations), no matter which notation we use to write down sucxh a triple. However, while we'd usually consider $\mathbb R^3$ as the set of triples of real numbers and write such triples in the form $(x,y,z)$, for the vector space $\mathbb R^3$, it is customary to use the column form, i.e. write vectors as in your form 1. The reason is that this makes most sense when introducing matrix multiplication (i.e., $Av$ where $A$ is a matrix and $v$ a vector). That would make a row vector a one-by-three matrix, an element of $\mathbb R^{1\times 3}$. But these two structures are isomorphic (i.e., essentiually indistinguishable).
• @ChrisAl In a vector space (over the real numbers, to be sure) where we have a notion of orthogonality by means of some scalar product (not all vector spaces have such a scalar product naturally!), we can in fact define angles between nonzero vectors via $\cos(v,w)=\frac{v\cdot w}{\sqrt{v\cdot v}\sqrt{w\cdot w}}$ or simply say that two vectors point in the same direction if they are obtained from each other by multipying with a (positive) constant. – Hagen von Eitzen May 15 '15 at 16:31