a cube has six faces and twelve edges. you start at one corner,are allowed to move only along the edges and may not retrace your path along any edge. Consistent with these rules, there are any number of ways to arrive back at your starting point. For instance, you could move around the four edges that make up one of the square faces. The four corresponding displacement vector swould add to zero. How many ways are there to arrive back at your starting point that involve eight displacement vectors that add to zero? Describe each possibility, using drawings for clarity.
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To solve this problem, we need to find all the possible ways to arrive back at the starting point on a cube using eight displacement vectors that add up to zero.
First, let's understand how many possible displacement vectors can add up to zero. Each displacement vector can have three possible directions (x, y, or z coordinate) and a negative or positive value.
Since we need eight vectors that add up to zero, we must have an even number of positive and negative vectors. Let's denote a positive vector as "+" and a negative vector as "-".
Now, let's consider the possibilities:
1. + + - - + + - -:
In this case, we have four positive and four negative vectors. One possible path is to move around a square face with four edges in one direction and then return along the same path but in the opposite direction.
Starting at a corner, we can move along the first edge (+), then the second edge (+), followed by the third edge (-), and finally the fourth edge (-). In this manner, we return to the starting point by reversing the directions.
Drawing:
[Corner] -- (+) -- (+) -- (-) -- (-) --> [Corner]
2. + + + - - - -:
In this case, we have three positive and five negative vectors. One possible path is to move along a triangular face with three edges in one direction and then return along two opposing edges.
Starting at a corner, we can move along the first edge (+), then the second edge (+), and the third edge (+) to reach an adjacent corner. From there, we can move along the fourth edge (-) to another adjacent corner and finally return along the fifth edge (-) to the starting point.
Drawing:
[Corner] -- (+) -- (+) -- (+)
\
\--> (-) -- (-) -- (-) --> [Corner]
3. + + - + - - +:
In this case, we have three positive and five negative vectors. One possible path is to move along a triangular face with three edges in one direction and then move along three edges of another square face in the opposite direction.
Starting at a corner, we can move along the first edge (+), then the second edge (+), and the third edge (-) to reach an adjacent corner. From there, we can move along the fourth edge (+) and fifth edge (-) to reach another adjacent corner. Finally, we move along the sixth edge (-) to return to the starting point.
Drawing:
[Corner] -- (+) -- (+) -- (-)
\
\--> (+) -- (-) -- (-) --> [Corner]
4. - + - + - + +:
In this case, we have four positive and four negative vectors. One possible path is to alternately move along the edges of two opposing square faces.
Starting at a corner, we can move along the first edge (-), then the second edge (+), followed by the third edge (-) to reach a corner of an opposing square face. From there, we move along the fourth edge (+), fifth edge (-), and sixth edge (+) to reach a corner of the initial square face. Finally, we move along the seventh and eighth edges (+) to return to the starting point.
Drawing:
[Corner] -- (-) -- (+) -- (-) -- (+)
\
\--> (-) -- (+) --> [Corner]
These are the four possible paths using eight displacement vectors that add up to zero on a cube.