Question

If c~ is a constant vector field and ~r stands for the position vector with respect to the origin, ~ ~ cr ~ 2 =?

Answer 1

Assume that by ar the OP means a⋅r, where r is the position vector. Then with a=(ax,ay,az)T and r=(x,y,z)T, we have

a⋅r=axx+ayy+azz;(1)

and since

∇(a⋅r)=(∂a⋅r∂x,∂a⋅r∂y,∂a⋅r∂z)T,(2)

we merely need observe that

∂a⋅r∂x=∂(axx+ayy+azz)∂x=ax,(3)

with the analogous results holding for partial differentiation with respect to y and z. Thus

∇(a⋅r)=(ax,ay,az)T,(4)

as was to be shown. QED.

Hope this helps.