Practicing Success
The edges of a parallelopiped are of unit length and are parallel to noncoplanar unit vectors $\displaystyle \hat{a}, \hat{b}, \hat{c}$ such that $\displaystyle \hat{a}\cdot \hat{b}= \hat{b}\cdot\hat{c}= \hat{c}\cdot \hat{a}= \frac{1}{2}$, then the volume of the parallelopiped is |
$\frac{1}{\sqrt{2}}$ $\frac{1}{2\sqrt{2}}$ $\frac{\sqrt{3}}{2}$ $\frac{1}{\sqrt{3}}$ |
$\frac{1}{\sqrt{2}}$ |
Let V be the volume of the parallelepiped. Then $V^2=|\hat a.(\hat b×\hat c)|^2=\begin{vmatrix}\hat a.\hat a&\hat a.\hat b&\hat a.\hat c\\\vec b.\vec a&\vec b.\vec b&\vec b.\vec c\\\vec c.\vec a&\vec c.\vec b&\vec c.\vec c\end{vmatrix}=\begin{vmatrix}1&1/2&1/2\\1/2&1&1/2\\1/2&1/2&1\end{vmatrix}=\frac{1}{2}$ Hence (A) is the correct answer. |