Understanding of the line passing through $u$ in the direction of $v$

Question

Definition. Let $u,v$ be vectors, $v\neq 0$. Then we define

$l_{u,v}$={$tv+u:t\in\mathbb{R}$}.

This is called the line passing through $u$ in the direction of $v$. A line is simply a set of the form $l_{u,v}$.

Can you give example? Because I didn't understand that what the definiton say.

Answer 1

Best if you draw any vector $v$ and call its startpoint $u$. Then it's about the line through $u$ in the direction of $v$.

A simple example with coordinates (draw it as well:)

• $u=(0,1)$ this is a point one unit above of the origin
• $v=(2,-1)$ this is a vector going $2$ to the right and $-1$ upwards, i.e. $1$ downwards.

The next whole coordinate points in the line through this $u$ in the direction of $v$ are $(2,0),\ (4,-1)$ to the right and $(-2,2),\ (-4,3)$ to the left.

The $t$ values for these points were $t=1,2,-1,-2$. This means that e.g. $(2,0)={\bf1}\cdot (2,-1) + (0,1)$, or $(-4,3)={\bf-2}\cdot (2,-1)+(0,1)$...