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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.
Solution
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
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