Coordinate Frames

My understanding of notation:

• When we write $[p]$, its like saying this is a set of signed ($-/+$) values that represents a physical point $p$.
• When we add the $X$ on top, $[p{]}^X$, its like saying this set represents signed values in the coordinate frame $X$.
• In a 3-Dimensional space where $R^3$, $[p{]}^X$ will give us something like $p=0.2x^1+1.5x^2+9.3x^3$
• We can clearly see the 3 dimensions. Remember that the $X$ above just indicates which coordinate frame point p is referring to.
• Another notation is when we use $[p{]}_1^X$, this means we wanna only look at the signed magnitude of point $p$'s $x^1$/$x_1$. In the above example, it is 0.2.

Imagine $p=0.2x^1+1.5x^2+9.3x^3$, and coordinate frame is $X=\{x_1,x_2,x_3\}$, the first coordinate is the signed projection of $p$ onto the first unit vector: $p_1^X=p\cdot x_1$.