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Math Help - Get the set of every direction of S

  1. #1
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    Get the set of every direction of S

    Well, d is a direction of a set S if for all x \in{S}, \lambda \geq{0}, x+\lambda d \in{S}.
    Let S=\left\{{(x_1,x_2): x_1-2x_2 \geq{-6}, x_1-x_2\geq{-2}, x_1\geq{0}, x_2 \geq{1}}\right\} 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 d=(d_1,d_2), d_1 = 2\alpha + \beta; d_2=\alpha: \alpha, \beta \geq{0}. Could anyone help me?
    Thanks.
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  2. #2
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    Re: Get the set of every direction of S

    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 x_2 = 1, bounded on the left by x_1 = 0 and bounded above by

    x_2 = x_1 + 2 and x_2 = \dfrac{1}{2}x_1 + 3.

    of these two lines, the second one has the lesser slope, and eventually is the more restrictive.

    so for (x_1,x_2) + \lambda(d_1,d_2) \in S, whenever (x_1,x_2) \in S for all \lambda > 0 we have to have:

    d_1 \geq 0,\ d_2 \leq \dfrac{1}{2}d_1

    that is, (d_1,d_2) must lie between the rays (\alpha,0) and (\alpha,\alpha/2), \alpha \geq 0.
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  3. #3
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    Re: Get the set of every direction of S

    Quote Originally Posted by Deveno View Post
    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 x_2 = 1, bounded on the left by x_1 = 0 and bounded above by

    x_2 = x_1 + 2 and x_2 = \dfrac{1}{2}x_1 + 3.

    of these two lines, the second one has the lesser slope, and eventually is the more restrictive.

    so for (x_1,x_2) + \lambda(d_1,d_2) \in S, whenever (x_1,x_2) \in S for all \lambda > 0 we have to have:

    d_1 \geq 0,\ d_2 \leq \dfrac{1}{2}d_1

    that is, (d_1,d_2) must lie between the rays (\alpha,0) and (\alpha,\alpha/2), \alpha \geq 0.
    Thanks for answering but I don't see how do you get those inequalities: d_1 \geq 0,\ d_2 \leq \dfrac{1}{2}d_1.

    Thank you again.
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  4. #4
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    Re: Get the set of every direction of S

    it might help if you draw a picture of S.

    what happens if we add (d_1,d_2) to the point (0,1) (which is in the lower left corner of S), if d_1 < 0? 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 d_2 \geq 0 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 \infty going up and to the right.

    but now, if the slope of our ray \lambda(d_1,d_2) is greater than 1/2, we'll eventually cross the line:

    x_2 = \frac{1}{2}x_1 + 3, which will take us outside of S. at best, we can only go parallel to it. this limits which directions we can go, for (d_1,d_2) to work,

    d_2 has to be less than or equal to 1/2 of d_1.

    it's a lot easier to understand all this, if you draw S, because then you can SEE it.
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