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Linear Algebra on a Regular Hexagon

  1. Jan 15, 2014 #1
    1. The problem statement, all variables and given/known data

    We are supposed to compute the magnitude of vectors that make up a regular hexagon. We are given the magnitude of one side (its magnitude is 1).

    We are also supposed to compute one of the interior angles.


    2. Relevant equations

    I feel like this isn't enough information to put into any equation I know.

    3. The attempt at a solution

    Obviously because it's a regular hexagon, all sides (vectors in the plane) are of equal length to each other.
    Also, I am aware of this fact: each time a side is added to a polygon, 180 degrees is added to the total of the interior angles. A triangle is 180 degrees, square is 360, pentagon is 540 and hexagons are 720, which would make each interior angle 120 degrees.

    But I believe the professor wants us to use linear algebra to show this, especially since the directive compute was given. I think my attempt at a solution is just deductive reasoning/ inference. Does anyone know of a way to use linear algebra to find these values? The hexagon has a vertex at the origin and one vector is aligned with the x axis, which means part of the hexagon is in the second quadrant. Any help would be greatly appreciated. Thanks in advance,

    Lee
     
    Last edited: Jan 15, 2014
  2. jcsd
  3. Jan 15, 2014 #2
    My guess is that by compute he means to compute the vector length from the origin to the various points on the hexagon. To do this you need a way of representing each point P on the perimeter of the hexagon. That representation will depend on the angle between the origin and P. It is perfectly okay if the angle is > 90° for the points in the 2nd quadrant.

    I think the place to start is by computing the location of the center of your hexagon. If you create a triangle using the side on the x-axis running from (0,0) to (1,0); and the radii from the center of the hexagon, what is the angle at the center of the hexagon? What is the angle at (0,0? what is the angle at (1,0)?

    From this information you should be able to use elementary geometry and trigonometry to calculate the location of the center; which in turn gets you the height of the hexagon.

    The next step is to compute the location of a point on the perimeter of your hexagon. This can be done directly and is probably what your prof intends. However, if it were me, I would move the center of the hexagon to the origin, which greatly simplifies the computations. ( I am no fan of computing more when there is an option to compute less.)

    You do this with a "translation". You are presently working in x,y coordinates. If the center of the hexagon is at (a,b), you set up new coordinates u,v where u = x -a and v = y -b. Eventually, you will translate the coordinates back to x,y but first you will compute all the points on the perimeter of your translated hexagon.

    Before I provide further guidance, can you do the steps above? That is, compute the center of the hexagon, and create the u,v axis?
     
  4. Jan 15, 2014 #3
    I appreciate the help, but my first sentence of the post was that we are computing the magnitude of the vectors that make up the hexagon, i.e the sides. I figured it out though.

    Lee
     
  5. Jan 15, 2014 #4
    Sorry I misunderstood. It was not entirely clear and I do not guess well. But the "magnitude" of the vector is its length isn't it? And the sides are defined to be of length 1. Thus I still do not understand the question.

    However, I'm glad you solved it.
     
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