# 3d representation of a regression with two independent variables one of them is categorical and another is continuous

I have hopefully a fundamental question of Do I understand things right. (Thank you in advance and sorry for my English which might be not so good)

1-Preambula 1: I know that if we have 2 independent variables, both of a continuous type, it is ok to represent them as a 2d plane in a 3d space:

2-Preambula 2: I have seen that many times when we have to deal with continuous and categorical variables(male\female for example), we represent them like this(note the lines are parallel):

3-Assumption: In the beginning I assumed that it is 2d representation of this 3d case: 4-Discussion 1: But If my assumption above was right, why do they always "picture" it with parallel lines? After all this is a very specific situation. In most of cases both regression lines will not be parallel, further more, they may have different slope direction (one negative and another positive)For example:

5-Discussion 2: On the other hand parallel models may be explained in such a way: if we will add a regression hyperplane which "somehow" fits both groups(male and female), we will get the parallel lines:

6-Finally My questions are quite simple.

question 5.1: Did I understand right the nature of parallel lines as I show it above (in discussion2)?

question 5.2: If I was right in 5.1, I assume that in such cases hyperplane regression is a quite a bad predictor. Am I right?