First slide

Scalar triple product of vectors

Question

If V be the volume of a tetrahedron and V¹ be the volume of another tetrahedron formed by the centroids of faces of the previous tetrahedron and V = KV¹ , then K is equal to

Moderate

A

9
B

12
C

27
D

81

Solution

Consider a tetrahedron with vertices O(0,0, 0), A(a,0, 0), B(0, b, 0) and C(0 , 0, c).
$Volume V = \frac{1}{6} [\vec{a} \vec{b} \vec{c}]$
Now centroids of the faces OAB, OAC, OBC and ABC arc G₁(a/3, b/3,0), G₂(3,0, c/3), G₃(0, b/3, c/3) and G₄(a/3, b/3, c/3), respectively.
$G_{4} G_{1} = \vec{c} / 3, {\vec{G_{4} G}}_{2} = \vec{b} / 3, {\vec{G_{4} G}}_{3} = \vec{a} / 3$
Volume of tetrahedron by centroids
$\begin{array}{l} V^{'} = \frac{1}{6} [\begin{array}{lll} \frac{a}{3} & \vec{b} & \frac{\vec{c}}{3} \end{array}] = \frac{1}{27} V \\ \Rightarrow K = 27 \end{array}$

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