The edges of a parallelopiped are of unit length and are parallel...

The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vectors

\hat{\mathbf{a}}, \hat{\mathbf{b}}, \hat{\mathbf{c}}

such that

\hat{\mathbf{a}} \cdot \hat{\mathbf{b}}=\hat{\mathbf{b}} \cdot \hat{\mathbf{c}}=\hat{\mathbf{c}} \cdot \hat{\mathbf{a}}=\frac{1}{2}

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

The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vectors a , b , c such that a ·

Options

A $\frac{1}{\sqrt{2}} \mathrm{cu}$ units
B $\frac{1}{2 \sqrt{2}}$ cu units
C $\frac{\sqrt{3}}{2}$ cu units
D $\frac{1}{\sqrt{3}} \mathrm{cu}$ units

Correct answer

A. $\frac{1}{\sqrt{2}} \mathrm{cu}$ units

Step-by-step solution

The volume of the parallelopiped with coterminus edges as

\hat{\mathbf{a}}, \hat{\mathbf{b}}, \hat{\mathbf{c}}

is given by

[\hat{\mathbf{a}} \hat{\mathbf{b}} \hat{\mathbf{c}}]=\hat{\mathbf{a}} \cdot(\hat{\mathbf{b}} \times \hat{\mathbf{c}})

\begin{aligned}& \text{ Now, }[\hat{\mathbf{a}} \hat{\mathbf{b}} \hat{\mathbf{c}}]^2=\left|\begin{array}{ccc} \hat{\mathbf{a}} \cdot \hat{\mathbf{a}}& \hat{\mathbf{a}} \cdot \hat{\mathbf{b}}& \hat{\mathbf{a}} \cdot \hat{\mathbf{c}}\\ \hat{\mathbf{b}} \cdot \hat{\mathbf{a}}& \hat{\mathbf{b}} \cdot \hat{\mathbf{b}}& \hat{\mathbf{b}} \cdot \hat{\mathbf{c}}\\ \hat{\mathbf{c}} \cdot \hat{\mathbf{a}}& \hat{\mathbf{c}} \cdot \hat{\mathbf{b}}& \hat{\mathbf{c}} \cdot \hat{\mathbf{c}} \end{array}\right|=\left|\begin{array}{ccc}1 & 1 / 2 & 1 / 2 \\ 1 / 2 & 1 & 1 / 2 \\ 1 / 2 & 1 / 2 & 1 \end{array}\right|=\frac{1}{2}\\ & \Rightarrow \quad[\hat{\mathbf{a}}\hat{\mathbf{b}} \hat{\mathbf{c}}]^2=\frac{1}{2}\\ & \end{aligned}

Thus,

the

required volume of the parallelopiped =\frac{1}{\sqrt{2}}\mathrm{cu} units$

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