Question

dimension of 3rd equation of motion

Answer 1

umm 3-d dude. acceleration, velocity and position are 3-d vectors (x, y, z) and their derivatives.

so v = ds/dt; a = dv/dt; but where s is a vector distance. Ahh I see the confusion, s.s is the scalar square of position; when we say s^2 = s.s position where s is a vector; however s.s is the dot product of two vectors and hence is scalar.

[ reminder: the dot product of vectors a.b = |a|.|b|.cos(theta); thus the dot product of vectors s.s = |s|.|s|.cos(0) = |s|^2; ]

Just replace the normal single law of inertia with 3-d vectors for now in normal Cartesian coordinates.