Question

Find the volume of the parallelopiped spanned by the diagonals of the three faces of a cube of side a that meet at one vertex of the cube.

Answer 1

Take origin O as one vertex of the cube and OA, OB and OC as the positive directions of the X-axis, the Y-axis and the Z-axis respectively. Here, the sides of the cube are 
OA = OB = OC = a

∴ the coordinates of all the vertices of the cube will be

O (0, 0, 0) B(0, a, 0) N(a, a, 0) M(a, 0, a) A(a, 0, 0) C(0, 0, a) L (0, a, a) P(a, a, a)

ON, OL, OM are the three diagonals which meet at the vertex O

`bar"ON" = bar"a"hat"i" + bar"a"hat"j", bar"OL" = bar"a"hat"j" + bar"a"hat"k"`

`bar"OM" = bar"a"hat"i" + bar"a"hat"k"`

`[bar"ON"  bar"OL"  bar"OM"] = |("a","a",0),(0,"a","a"),("a",0,"a")|`

= a(a² - 0) -a(0 - a²) + 0

= a³ + a³ = 2a³

∴ required volume = `[bar"ON" bar"OL" bar"OM"]`

= 2a³ cubic units