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I am studying "Introduction to Multivariable Mathematics (page 13)", and I came across the following line in the section "Dot product and angle between vectors":

...for $\overline{y} \neq 0$, the straight line $\overline{l}$ through $\overline{x}$ parallel to $\overline{y}$ is by definition the set of points $\{\overline{x} + t\overline{y} : t ∈ \mathbb{R}\}$.

I wanted to make sure $\overline{l}$ is indeed passing through $\overline{x}$ and is parallel to $\overline{y}$, but I am having some trouble.

I tried to come up with examples of points but I ended up with a line NOT parallel to $\overline{y}$.

This is an example I tried:

$\overline{x} = \{(0, 1), (1, 3), (2, 5)\}$

$\overline{y} = \{(1, 2), (3, 3), (5, 4)\}$

$t = 2$

hence,

$\overline{l} = \{(2, 5), (7, 9), (12, 13)\}$

As you can see, $\overline{l}$ is not parallel to $\overline{y}$.

My understanding of the statement is wrong, or my example is flawed or both. I'd like to understand this concept. Thanks!

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The value that you got is just a single point on the line. In order to get the vector that's parallel to the line's direction you must subtract two points.

Infact it's very easy to see that for any two points on the line $\bar l_0 = \bar x + t_0 \bar y $ and $\bar l_1 = \bar x + t_1 \bar y $ their difference is scalar multiple of $\bar y$ hence it's parallel to it $$ \bar l_1 - \bar l_0 = (t_1 - t_0) \bar y $$

You can think of $l_1$ and $l_0$ as directions from the origin to that point on your line. Those directions won't be parallel to your line.

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