Dealing with vectors, we usually follow the two following conventions:
CONVENTION $\boldsymbol 1$:
Vectors are denoted with an arrow; their magnitude is denoted by the same letter without the arrow.
For example, $\vec v$ is a vector and $v$ is its magnitude.
CONVENTION $\boldsymbol 2$:
The vector components of the bidimensional vector $\vec v$ are denoted by $\vec v_x$ and $\vec v_y$. The scalar components of the bidimensional vector $\vec v$ are denoted by $v_x$ and $v_y$.
The vector components are the only two vectors, one parallel to the $x$ axis and one parallel to the $y$ axis, such that $\vec v = \vec v_x + \vec v_y$:
The scalar components are the only two numbers such that $\vec v = v_x \hat x + v_y \hat y$, where $\hat x$ and $\hat y$ are the unit vectors in the direction of the $x$ axis and the $y$ axis, respectively.
These two conventions are inconsistent because, based on convention $1$, one would expect $v_x$ to be the magnitude of $\vec v_x$, which implies that $v_x \geq 0$. However, based on convention $2$, $v_x <0$ for the vector represented in the drawing.
There isn’t any major conceptual difficulty here, but I keep noticing that the high school students I tutor find this extremely confusing. Do any of you know of some nice notation for vector components and scalar components that avoids this issue?
