Question

If the volume of the parallelepiped whose conterminus edges are along the vectors a, b, c is 12, then the volume of the tetrahedron whose conterminus edges are a + b, b + c and c + a is ______.

विकल्प

• 12 (units)³ 

• 24 (units)³

• 4 (units)³

• 6 (units)³

Answer 1

If the volume of the parallelepiped whose conterminus edges are along the vectors a, b, c is 12, then the volume of the tetrahedron whose conterminus edges are a + b, b + c and c + a is 4 (units)³.

Explanation:

We have,

a, b, care coterminus edges of parallelepiped

∴ volume of parallelepiped = [a · b · c]

a + b, b + c and c + a are coterminus edges of tetrahedron

∴ Volume of tetrahedron =`1/6`[a+b b+c c+a]

`=1/6xx2["a b c"]`  [∵ [a+b b+c c+a] = 2[a b c]

`=1/3["a b c"]=12/3=4`   [∵[a b c] = 12]

∴ Volume of tetrahedron = 4 (unit)³