# Confusion with parallel vector definition.

Two non-zero vectors $x$ and $y$ are said to be parallel if there exists a non-zero scalar $\lambda$ such that $x=\lambda y$.

Here is my stupid question: I don't understand this definition in $\mathbb{R^2}$ or $\mathbb{R^3}$. If $x$ is a scalar multiple of $y$, then how are they parallel? In $\mathbb{R^2}$, isn't $y$ contained in $x$ since they're scalar multiples? So they're lines with direction, but they share part of the same line. So how are they parallel if they have points in common with each other? I'm thinking the same thing in $\mathbb{R^3}$.

• A vector is not a line. For each vector, there are an infinite many number of directed line segments that represent that vector. For any vector $\vec V$, we can create a directed line segment that begins at a point, say $(x_0,y_0,z_0)$, and ends at another point, $(x_1,y_1,z_1)=(x_0,y_0,z_0)+1\,\vec V$. Commented Jul 18, 2015 at 4:04
• I don't believe this definition is correct, as the zero vector is parallel to all vectors even though it doesn't meet your definition Commented Dec 28, 2022 at 23:41

I imagine you are thinking of the geometric definition of parallel lines, which is that two lines in $\mathbb R^n$ are parallel if they do not intersect, or if they intersect an infinite number of times. Since vectors are not lines, this definition does not apply, and we use a new definition, which is the one you gave.