In his book on Engineering Mechanics - Statics, R C Hibbeler provides many force problem solutions in both scalar and Cartesian notation (e.g Example 2.5 Chapter 2).
It feels like he is trying to articulate some significant difference between the two notations and writes;
'Comparing the two methods of solution, notice that the use of scalar notation is more efficient since the components can be found directly, without first having to express each force as a Cartesian vector before adding the components. Later, however, we will show that Cartesian vector analysis is very beneficial for solving three-dimensional problems'.
It seems to me like the only difference between the two methods of describing the force vectors is a swap from some method of denoting the axis direction for each component (scalar notation) to the unit vector i,j or k (Cartesian). It seems quite superficial. I don't understand how it is fundamentally different to the scalar method of finding the resultant of a system of forces and therefore suddenly very beneficial for solving three dimensional problems. I understand there are relationships such as those involving the direction Cosines which are perhaps neater if expressed with the Cartesian unit vectors - but I still fundamentally see no real difference that makes everything easier.
Am I missing something fundamental?
Many Thanks
You are correct, the approaches are identical. However, the "cartesian vector notation" as he calls it, is just a step towards abstraction that lets you write things simpler.
As an example, say you wanted to write $F=ma$ in three dimensions.
You could write $$F_{x}=ma_{x},F_{y}=ma_{y},F_{z}=ma_{z}$$
Or you could write $$\vec{F}=m\vec{a}$$
Where $\vec{F}=(F_{x},F_{y},F_{z})$, and $\vec{a}=(a_{x},a_{y},a_{z})$.