The geometric meaning of a line plus a vector Lets say we have
$$
E = \{k(1,2,3)' + (2,9,-1)'\} \;\mathrm{with}\; k \in \mathbb{R}
$$
we know that $k(1,2,3)$ spans a line in three dimensions, but what does the shape of $E$ look like. I think it is an infinite rectangle but only in one direction, namely along the line
$k(1, 2, 3)'$. Am I right? Thank you very much.

The set $\{k(1, 2, 3) \mid k \in \mathbb{R}^3\}$ is the line in $\mathbb{R}^3$ through the origin in the direction of the vector $(1, 2, 3)$. 
The set $\{k(1, 2, 3) + (2, 9, -1) \mid k \in \mathbb{R}^3\}$ is the line in $\mathbb{R}^3$ through the point $(2, 9, -1)$ in the direction of the vector $(1, 2, 3)$. All we've done is translated the entire line $\{k(1, 2, 3) \mid k \in \mathbb{R}^3\}$ by the vector $(2, 9, -1)$.

Consider the two-dimensional analogue of this question. What is the difference between $\{k(1, 2) \mid k \in \mathbb{R}\}$ and $\{k(1, 2) + (2, 9) \mid k \in \mathbb{R}\}$? The former is the line through the origin in the direction of the vector $(1, 2)$, while the latter is the line through the point $(2, 9)$ in the direction of the vector $(1, 2)$. This can be observed in the image below where the first line is the blue one.
$\hspace{45mm}$
