The edges of a parallelepiped are of unit len | vedclass

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(A) The volume of a parallelepiped defined by vectors $\hat{a}, \hat{b}, \hat{c}$ is given by the scalar triple product $|\hat{a} \cdot (\hat{b} 	ime

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$\frac{1}{\sqrt{2}}$

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(A) The volume of a parallelepiped defined by vectors $\hat{a}, \hat{b}, \hat{c}$ is given by the scalar triple product $|\hat{a} \cdot (\hat{b} 	imes \hat{c})|$.This is equal to the square root of the determinant of the Gram matrix:Volume $= \sqrt{\det \begin{bmatrix} \hat{a} \cdot \hat{a} & \hat{a} \cdot \hat{b} & \hat{a} \cdot \hat{c} \ \hat{b} \cdot \hat{a} & \hat{b} \cdot \hat{b} & \hat{b} \cdot \hat{c} \ \hat{c} \cdot \hat{a} & \hat{c} \cdot \hat{b} & \hat{c} \cdot \hat{c} \end{bmatrix}}$.Given $\hat{a} \cdot \hat{a} = \hat{b} \cdot \hat{b} = \hat{c} \cdot \hat{c} = 1$ and $\hat{a} \cdot \hat{b} = \hat{b} \cdot \hat{c} = \hat{c} \cdot \hat{a} = 1/2$,we have:Volume $= \sqrt{\det \begin{bmatrix} 1 & 1/2 & 1/2 \ 1/2 & 1 & 1/2 \ 1/2 & 1/2 & 1 \end{bmatrix}}$.Calculating the determinant: $1(1 - 1/4) - 1/2(1/2 - 1/4) + 1/2(1/4 - 1/2) = 1(3/4) - 1/2(1/4) + 1/2(-1/4) = 3/4 - 1/8 - 1/8 = 3/4 - 2/8 = 3/4 - 1/4 = 1/2$.Thus,the volume is $\sqrt{1/2} = \frac{1}{\sqrt{2}}$.

The edges of a parallelepiped are of unit length and are parallel to non-coplanar unit vectors $\hat{a}, \hat{b}, \hat{c}$ such that $\hat{a} \cdot \hat{b} = \hat{b} \cdot \hat{c} = \hat{c} \cdot \hat{a} = 1/2$. Then the volume of the parallelepiped is

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