Advertisement Remove all ads

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. - Mathematics and Statistics

Advertisement Remove all ads
Advertisement Remove all ads
Sum

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.

Advertisement Remove all ads

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 

Concept: Vectors and Their Types
  Is there an error in this question or solution?
Advertisement Remove all ads

APPEARS IN

Advertisement Remove all ads
Share
Notifications

View all notifications


      Forgot password?
View in app×