A cube of side 5 has one vertex at the point (1, 0, –1), and the three edges from this vertex are, respectively, parallel to the negative x and y axes and positive z-axis. Find the coordinates of the other vertices of the cube.

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Solution

Let P $\equiv$ (1, 0, −1)
The length of each side of the cube is 5.
The three edges from vertex of the cube are drawn from P towards the negative x and y axes and the positive z-axis.
Therefore, the coordinates of the vertex of the cube will be as follows:
x-coordinate = 1, 1 $-$ 5 = $-$ 4, i.e. 1, $-$ 4
y-coordinate = 0, 0 $-$ 5 = $-$ 5, i.e. 0, $-$ 5
z-coordinate = $-$ 1, $-$ 1 + 5 = 4, i.e. $-$ 1, 4
Hence, the remaining seven vertices of the cube are as follows:
$(1, 0, 4) (1, - 5, - 1) (1, - 5, 4) (- 4, 0, - 1) (- 4, 0, 4) (- 4, - 5, - 1) (- 4, - 5, 4)$

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