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IIT-JEE 2008 · Paper 1, Mathematics question 3

The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vectors a^,b^,c^\hat a,\hat b,\hat c such that
a^⋅b^=b^⋅c^=c^⋅a^=12.\hat a\cdot\hat b=\hat b\cdot\hat c=\hat c\cdot\hat a=\frac12.
Then, the volume of the parallelopiped is

Choose one option
Answer and worked solution

Official answer: A

Volume squared equals the Gram determinant of the edge vectors:
V2=∣112121211212121∣=1−3⋅14+2⋅18=12.V^2=\begin{vmatrix}1&\frac12&\frac12\\\frac12&1&\frac12\\\frac12&\frac12&1\end{vmatrix}=1-3\cdot\frac14+2\cdot\frac18=\frac12.

Volume. V=12V=\dfrac1{\sqrt2}.

(A)

Topics:Scalar and vector triple products
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