Well, d is a direction of a set S if for all .
Let I have to get the set of every direction of S.
And I have the answer but I don't know how to reach it. The answer is: d has two components . Could anyone help me?
Thanks.
Well, d is a direction of a set S if for all .
Let I have to get the set of every direction of S.
And I have the answer but I don't know how to reach it. The answer is: d has two components . Could anyone help me?
Thanks.
a direction is not a number. it could be a vector (like say, a unit vector).
your set S is a subset of the first quadrant. it is bounded below by , bounded on the left by and bounded above by
and .
of these two lines, the second one has the lesser slope, and eventually is the more restrictive.
so for , whenever for all we have to have:
that is, must lie between the rays and .
it might help if you draw a picture of S.
what happens if we add to the point (0,1) (which is in the lower left corner of S), if ? isn't the first coordinate going to shift to the left? that will put us outside of S. so we know we can only move to the right.
that leaves up-and-down. now, i should have included the condition as well, because if we go down, the second coordinate could drop below 1, and that will cause us to leave S as well.
so, we can only go up and to the right. well that's 3/4ths of the possible directions eliminated. you can think of this as all slopes between 0 and going up and to the right.
but now, if the slope of our ray is greater than 1/2, we'll eventually cross the line:
, which will take us outside of S. at best, we can only go parallel to it. this limits which directions we can go, for to work,
has to be less than or equal to 1/2 of .
it's a lot easier to understand all this, if you draw S, because then you can SEE it.