miller index planes and directions relationship

[cameuth]

1. The problem statement, all variables and given/known data
do you notice a relationship between the plane and directions of the same miller index? what is it?


2. Relevant equations

I've done planes and directions (111), [111], (112), and [112]

3. The attempt at a solution
I believe the direction is normal to the plane on first glance, but I can't seem to do the math to prove it. first off, am I right that the direction is perpendicular and second, how would I begin to prove this?

[Shayes]

yea the direction does appear to be perpendicular to the plane.
are you sure you need to prove it using math?

[cameuth]

I suppose I wouldn't have to prove it using math, but I do feel like I need some reason other than "they look perpendicular". I guess the reasoning behind why this happens is what I'm after.

[Shayes]

the way the question is worded implies you probably dont need to show using math.

is this an intro to materials class? or upper div?

[cameuth]

It's an upper level engineering class, but it's also a basic materials class. Basically, a lot is expected of us, so I want to be sure that I'm not missing some simple proof. If that were the case, I'd receive little to no credit for my answer.