The translation of a Cartesian coordinate system displaces the origin of the
original coordinates (x, y) by a vector (dx, dy) to a new position in a new coordinate
system (f, n). The transformation rule is given by
x=f+dx
y=n+dy
Consider two vectors in the original coordinate system given by a = (a1,a2) and
b = (b1 , b2). Show that the norm of a is not invariant under translation, but that
the norm of the vector a
Solution
as the origin change a and b varies because they are calculated with respect to the origin but a-b
is independent of placement of origin because it is not calculated with respect to origin but
vectors a and b so vector a-b is invariant,
and it will remain invariant in any other cartesian coordinates too
normally, a-b = (a1-b1 , a2-b2)
a=(f + da1 , n + da2)
b=(f + db1 , n + db2)

a-b = ((f + da1) - (f + db1) , (n + da2 ) - (n + db2))
= (d(a1-b1) , d(a2-b2))
it is independent on f and n,that's what i was telling that they are invariant and does not depend
on the change of origin