The document discusses how translating the origin of a Cartesian coordinate system affects vectors in that system. It states that while the norm of an individual vector changes with the translation of the origin, the norm of the difference between two vectors (a - b) remains invariant, as it does not depend on the placement of the origin but only on the difference between the vectors. It provides the transformation rules to show that the components of (a - b) are independent of the new origin coordinates f and n, demonstrating that the norm of (a - b) does not change with the translation.

1. 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)

2. 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