Question

The volume of tetrahedron whose adjacent edges are represented by the vectors $\overline{a},\overline{b},\overline{c}$

• A. $\frac{16\sqrt{2}}{3}$
• B. $8\sqrt{2}$
• C. $\frac{8\sqrt{3}}{3}$
• D. $\frac{32}{3}$

Answer 1

The correct option is A $\frac{16\sqrt{2}}{3}$

Given $\left|\overline{a}\right|=\left|\overline{b}\right|=\left|\overline{c}\right|=4or{\left|\overline{a}\right|}^2={\left|\overline{b}\right|}^2={\left|\overline{c}\right|}^2=16$
and $\overline{a}\cdot \overline{b}=\overline{b}\cdot \overline{c}=\overline{c}\cdot \overline{a}=8$
$\therefore \cos {\theta }_1=\cos {\theta }_2=\cos {\theta }_3=\frac{1}{2}$
$\Rightarrow {\theta }_1={\theta }_2={\theta }_3=\frac{\pi }{3}$
and $\overline{a}\cdot \overline{a}=\overline{b}\cdot \overline{b}=\overline{c}\cdot \overline{c}=16,{\overline{a}}^2={\overline{b}}^2={\overline{c}}^2=16$
Also $\left[\overline{a}\overline{b}\overline{c}\right]\left[\overline{x}\overline{y}\overline{z}\right]=\left|\begin{array}{ccc}\overline{a}\cdot \overline{x}& \overline{b}\cdot \overline{x}& \overline{c}\cdot \overline{x}\\ \overline{a}\cdot \overline{y}& \overline{b}\cdot \overline{y}& \overline{c}\cdot \overline{y}\\ \overline{a}\cdot \overline{z}& \overline{b}\cdot \overline{z}& \overline{c}\cdot \overline{z}\end{array}\right|$
${\left[\overline{a}\overline{b}\overline{c}\right]}^2=\left[\overline{a}\overline{b}\overline{c}\right]\left[\overline{a}\overline{b}\overline{c}\right]=\left|\begin{array}{ccc}\overline{a}\cdot \overline{a}& \overline{b}\cdot \overline{a}& \overline{c}\cdot \overline{a}\\ \overline{a}\cdot \overline{b}& \overline{b}\cdot \overline{b}& \overline{c}\cdot \overline{b}\\ \overline{a}\cdot \overline{c}& \overline{b}\cdot \overline{c}& \overline{c}\cdot \overline{c}\end{array}\right|$
$\Rightarrow {\left[\overline{a}\overline{b}\overline{c}\right]}^2=\left|\begin{array}{ccc}16& 8& 8\\ 8& 16& 8\\ 8& 8& 16\end{array}\right|=512\left|\begin{array}{ccc}2& 1& 1\\ 1& 2& 1\\ 1& 1& 2\end{array}\right|=1024\times 2$
$\left|\left[\overline{a}\overline{b}\overline{c}\right]\right|=32\sqrt{2}$ ......( * )
Volume of tetrahedron $=\frac{1}{6}$ volume of parallelopipedd
$=\frac{1}{6}\times 32\sqrt{2}=\frac{16\sqrt{2}}{3}$
$\therefore$ Choice (A) is correct option.