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Mathematics Question on Vectors

The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vector a^,b^,c^\widehat{a},\widehat{b},\widehat{c} such that a^.b^=b^.c^=c^.a^=12.\widehat{a}.\widehat{b}=\widehat{b}.\widehat{c}=\widehat{c}.\widehat{a}=\frac{1}{2}.Then, the volume of the parallelopiped is

A

12\frac{1}{\sqrt{2}}

B

122\frac{1}{2\sqrt{2}}

C

32\frac{\sqrt{3}}{2}

D

13\frac{1}{\sqrt{3}}

Answer

12\frac{1}{\sqrt{2}}

Explanation

Solution

The volume of the parallelopiped with coterminus edges
as a^,b^,c^\widehat{a},\widehat{b},\widehat{c} is given by [a^,b^,c^]=a^.(b^×c^)[\widehat{a},\widehat{b},\widehat{c}]=\widehat{a}.(\widehat{b}\times\widehat{c})
Now, [a^.b^c^]2=a^.a^a^.b^a^.c^b^.a^b^.b^b^.c^c^.a^c^.b^c^.c^=11/21/2\1/211/2\1/21/21Now,\ [\widehat{a}.\widehat{b}\widehat{c}]^2=\begin{array}| \widehat{a}.\widehat{a}&\widehat{a}.\widehat{b}&\widehat{a}.\widehat{c}\\\\\widehat{b}.\widehat{a}&\widehat{b}.\widehat{b}&\widehat{b}.\widehat{c}\\\\\widehat{c}.\widehat{a}&\widehat{c}.\widehat{b}&\widehat{c}.\widehat{c}\\\\\end{array}=\begin{array}|1&1/2&1/2\\\1/2&1&1/2\\\1/2&1/2&1\\\\\end{array}
⇒      [a^.b^c^]2=1(1−14)−12(1212)+12(1412)=12\Rightarrow \ \ \ \ \ \ [\widehat{a}.\widehat{b}\widehat{c}]^2= 1\Bigg(1-\frac{1}{4}\Bigg)-\frac{1}{2}\Bigg(\frac{1}{2}\frac{1}{2}\Bigg)+\frac{1}{2}\Bigg(\frac{1}{4}\frac{1}{2}\Bigg)=\frac{1}{2}
Thus, the required volume of the parallelopiped
=12cuunit=\frac{1}{\sqrt{2}} cu unit