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 $\mathbb{R^3}$.
Best Answer
A vector, in an elementary sense, is a magnitude and a direction. Two vectors are parallel if they have the same direction. Vectors are not lines or line segments. They are not sets, and therefore cannot contain one another. Vectors also are not points in the plane, nor do they contain points in the plane. This may seem strange to you as a vector is often represented as an arrow drawn on the plane, but a vector is actually a more abstract object.
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.