complex condition on dplyr mutate

Sample data:

library(dplyr)

id <- rep(LETTERS[1:5], each = 10)
x <- round(runif(50, -500, 200), digits = 0)
y <- round(runif(50, -700, 700), digits = 0)
z <- round(runif(50, 250, 300), digits = 0)

df.1 <- data.frame(id = id, x = x, y = y, z = z)
> summary(df.1)
 id           x                y                 z        
 A:10   Min.   :-497.0   Min.   :-665.00   Min.   :251.0  
 B:10   1st Qu.:-283.2   1st Qu.:-349.50   1st Qu.:261.2  
 C:10   Median :-128.0   Median : -33.50   Median :274.5  
 D:10   Mean   :-145.4   Mean   : -39.58   Mean   :275.3  
 E:10   3rd Qu.: -15.0   3rd Qu.: 293.25   3rd Qu.:288.0  
        Max.   : 171.0   Max.   : 696.00   Max.   :299.0  

What I'm trying to achieve are:

1. put each id into its own dataframe
2. create a new column called "direction" which would be a response to conditions below

a - identify the column with the widest range among x, y, z b - within the identified column, calculate direction by whether the next row value is bigger than the current row value - TRUE and FALSE return

i.e where y has the maximum range

  id    x    y   z direction
1  A -320   31 251      TRUE
2  A -199 -530 276     FALSE
3  A -228  390 264      TRUE
4  A -158  363 268      TRUE
5  A -308  150 267     FALSE
6  A  -47  345 261        NA

It is really important that the direction is calculated on the column that has the maximum range. In the sample data, it's likely column y is always the one with the maximum range, but in my real data it could be any column..

I imagine it'd involve mutate and ifelse?! but not sure how I'd go about it... I normally would use extensive for loop and only started using dplyr last week or two.. Trying not to fall back on to messy for loop and severely nested codes..

Appreciate your help very much! Thanks!

    for (i in 1:length(unique(id)) {

    x <- 
      df.1 %>% 
      filter(id == unique(id)[i] %>%
      mutate(direction = ifelse())

    assign(unique(id)[i], x)

      }