The edges of a parallelopied are of unit length and are parallel to non-coplanar unit vectors a¯,b¯,c¯ such that a¯b¯=b¯·c¯=c¯·a¯=12 . Then the volume of the parallelopiped is

a¯ b¯ c¯2=a¯.a¯a¯.b¯a¯.c¯b¯.a¯b¯.b¯b¯,c¯c¯.a¯c¯.b¯c¯.c¯= 112121211212121=12 Volume of parallel piped =[a¯b¯c¯]=12

The edges of a parallelopied are of unit length and are parallel to non-coplanar unit vectors a¯,b¯,c¯ such that a¯b¯=b¯·c¯=c¯·a¯=12 . Then the volume of the parallelopiped is

a¯ b¯ c¯2=a¯.a¯a¯.b¯a¯.c¯b¯.a¯b¯.b¯b¯,c¯c¯.a¯c¯.b¯c¯.c¯= 112121211212121=12 Volume of parallel piped =[a¯b¯c¯]=12

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

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

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

$\frac{\sqrt{3}}{2} cu units$

$\frac{1}{\sqrt{3}} cu units$

answer is A.

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${[\bar{a} \bar{b} \bar{c}]}^{2} = |\begin{matrix} \bar{a} . \bar{a} & \bar{a} . \bar{b} & \bar{a} . \bar{c} \\ \bar{b} . \bar{a} & \bar{b} . \bar{b} & \bar{b}, \bar{c} \\ \bar{c} . \bar{a} & \bar{c} . \bar{b} & \bar{c} . \bar{c} \end{matrix}|$

= $|\begin{matrix} 1 & \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & 1 & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} & 1 \end{matrix}| = \frac{1}{2}$

$Volume of parallel piped = [\bar{a} \bar{b} \bar{c}] = \frac{1}{\sqrt{2}}$

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The edges of a parallelopied are of unit length and are parallel to non-coplanar unit vectors a¯,b¯,c¯ such that a¯b¯=b¯·c¯=c¯·a¯=12 . Then the volume of the parallelopiped is

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