For example, $\vec{A}_{x}$ can signify $-5\hat{\imath}$. That for me is clear. However, $A_x$ seems to be a signed variable (-5) in some cases and an unsigned one (5) in others. This has caused a great deal of confusion for me. Gravitaional force is usually written as $-F_g$, so $F_g$ is unsigned, but then I see stuff like $F_g \hat{\imath}$ with $\hat{\imath}$ pointing upward, which must mean that $F_g$ is containing the negative.
Personal experience tells me that it's better for $A_x$ to be unsigned so that I can put the negatives when I need to, especially when working with multiple vectors, as in $(A_x – B_x – C_x)\hat{\imath}= \vec{D_x}$. But using this rule makes denoting the scalar part of the vector cumbersome, having to write $\vec{A_x} \cdot \hat{\imath}$ for just -5.
What notation rule should I use? Is there no symbol specifically for the "scalar part" of the vector component? The only solution I see is using absolute value signs everywhere, which gets cumbersome quick and is very easy to forget.
Best Answer
If I have a vector $\vec{A}=[-1,2,-3]$, I write the scalar components as $A_x = -1$, $A_y = 2$, and $A_z = -3$. I never work with unsigned values. Working with components this way allows for consistent writing of vector formulas. For example, the dot product $$A \cdot B = A_xB_x + A_yB_y + A_zB_z$$ or the cross product $$A \times B = \begin{bmatrix} A_yB_z - A_zB_y \\ A_zB_x - A_xB_z \\ A_xB_y - A_yB_x \end{bmatrix}.$$ There's no need to think about the signs of the scalar components when writing these formulas. The formula is always the same; just plug in the signed values.
This is similar to non-vector formulas. The position of an object that starts at $x_0$ and moves at a constant velocity $v$ for time $t$ is $x = x_0 + vt$. Do I care if the velocity has a positive or negative value? No. If the velocity is -5 m/s, then plug that in. Using $x_0 = 4 m$ and $t = 2 s$, we get $x = (4) + (-5)(2) = -6$. The object was moving in the negative direction as indicated by the negative velocity, so the formula results in a more negative position (however that is defined: down, left, etc.).