3/[2;2] gives
1×2 LinearAlgebra.Transpose{Float64,Array{Float64,1}}:
0.75 0.75
while 3 ./[2;2] gives
2-element Array{Float64,1}:
1.5
1.5
The second one is easy to comprehend. It broadcasts 3 and performs element wise division. But what is the reasoning behind having the first operation behave as it did? I assume it took the sum of the vector, which was 2x1, performed division of 3 by 4 and broadcast it to a 1x2 transposed vector. I can accept taking the sum of the vector to perform division, but why the transpose? Or why not just return a scalar?
It simply gives the right hand side operand's pseudo-inverse.
julia> ?/
...
Right division operator: multiplication of x by the inverse of y on the right.
Although it seems surprising at first sight, it is actually the natural behavior. A rowvector*columnvector gives a scalar and hence a scalar divided by a column vector should give a row vector, which is the case. Note that RowVector has been removed in 1.0 and what you get is actually a row vector represented with Transpose.
You can write @less 1 / [2;2] to see what actually happens.
Also take a look at this GitHub issue to understand the behaviour a bit more and this discourse topic for some use cases.