Creating a vector of values to match a distribution

Let's say that I have a vector $$u$$ of real values (say of length 100). I know that $$u-v$$ is normally distributed with mean $$\mu$$ and variance $$\sigma^2$$. I would like to calculate a vector of corresponding $$v$$ values.

So what I'm doing is like this: use $$rnorm()$$ to sample 10000 random values with mean $$\mu$$ and variance $$\sigma^2$$. Then for every value $$u_i$$ in $$u$$, I check which quantile $$u_i$$ is and then pick the value with the same quantile from the randomly generated values:

gen = rnorm(10000, mean=mu, sd=sigma)
for(i in 1:length(u)){
q = sum( u <= u[i] )/length(u)
v[i] = u[i] - quantile(gen, probs=q)
}


Is this correct? Are there any better ways to do it?

Is there a specific term for this procedure?