3D Geometry - Direction Cosine problem on cube

The vertices of a cube are (0,0,0) (0,0,1 ), (0,1,0), (1,0,0) (1,1,0) , (1, 0,1),(0,1,1) and (1,1,1) An insect starts from the origin and reaches the vertex (1,1,1) by crawling on the surface of the cube such that it travels the minimum distance Find the direction cosines of all the possible line segments of all the possible paths.


For minimum distance , i think this should be the path . Now you can write direction cosine easily !

Intuitively, you can come to the conclusion that the path taken by the insect would be the line joining the vertex and the midpoint of the opposite face on two adjacent faces (lengths would be equal, and you have to minimize the sum of the lengths travelled on adjacent faces)

From here on, finding the direction cosines is pretty straightforward, as you can find the coordinates of the points.

Note that there are 3 different paths of equal and shortest lengths.

There seem to be 6 in number. Two along each plane

@Amritaansh_2020 yes, there will be 6 , as there are 3 paths and there are 2 unique cosines for each path

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Yes sorry, its 6.