How do I normalize a vector?
user
For a vector A in a n dimensional space (a, b, c, …), the corresponding normalized vector is u = (a/z, b/z, c/z, …) where z = (a^2 + b^2 + c^2 …)^(1/2).

As an example in a 2 dimensional space:
Let vector A be a vector with its initial point at the origin and terminal point at (2,3), such that A = (2,3). Calculate the unit vector u = (x/(x^2 + y^2)^(1/2), y/(x^2 + y^2)^(1/2)) = (2/(2^2 + 3^2)^(1/2), 3/(2^2 + 3^2)^(1/2)) = (2/(13^(1/2)), 3/(13^(1/2))). Therefore, A = (2,3) normalizes to u = (2/(13^(1/2)), 3/(13^(1/2))).
assistant
Can you use an example for which the normalize vector does not contain any fraction?
user
Yes, here is an example where the normalize vector does not contain any fraction.

Consider a 3 dimensional vector A = (3,4,5). To normalize it, calculate the unit vector u = (x/z, y/z, z/z), where z = (x^2 + y^2 + z^2)^(1/2) = (3^2 + 4^2 + 5^2)^(1/2) = (9 + 16 + 25)^(1/2) = (50)^(1/2) = 7. So, the normalized vector is u = (3/7, 4/7, 5/7).

We can see that by multiplying each component of the normalized vector u by 7, we get back the original vector A.
assistant