"binning" rows into ranges (dplyr/R)

I am having a lot of difficulty trying to place rows from a dataset into "bins". For example, suppose I have a data frame "df" with "var1" and "var2" :

I want to create a new variable called "var3" that follows this logic (R code):

1) if var1 <5 and var2<5 .... then var3 = "a"
2) if var1 between (5,10) and var2 between (5,10) .... then var3 = "b"
3) if var1 > 10 and and var2>10 .... then var3 = "c"

From a previous question I posted (If statements with multiple ranges (R)), I tried the following logic:

library(dplyr)
df %>%
  mutate(var3 = case_when(var1 < 5 & var2 < 5 ~ 'a', 
                          var1 > 5 & var1 < 10 & var2 > 5 & var2 < 10 ~ 'b', 
                          var1 >10 & var2 >10 ~ 'c'))

But when I inspect the df$var3, the logic does not seem to be correct (i.e. some entries for var3 do not have any values. note: the smallest possible value of var1 and va2 is 0).

Can someone please help me?

Thanks

UPDATE:

Sample dataset:

a <- rnorm(50,10,10)
b <- rnorm(50, 2,8)

var1 = abs(a)
var2 = abs(b)

df = data.frame(var1, var2)