Question

$\left|\begin{array}{ccc}\overline{a}\cdot \overline{a}& \overline{a}\cdot \overline{b}& \overline{a}\cdot \overline{c}\\ \overline{b}\cdot \overline{a}& \overline{b}\cdot \overline{b}& \overline{b}\cdot \overline{c}\\ \overline{c}\cdot \overline{a}& \overline{c}\cdot \overline{b}& \overline{c}\cdot \overline{c}\end{array}\right|$ is equal to

• A. ${\left[\overline{a}\overline{b}\overline{c}\right]}^3$
• B. ${\left[\overline{a}\overline{b}\overline{c}\right]}^2$
• C. $\left[\overline{a}\overline{b}\overline{c}\right]$
• D. None of these

Answer 1

The correct option is B ${\left[\overline{a}\overline{b}\overline{c}\right]}^2$

Let

$\overline{a}={\alpha }_1\hat{i}+{\alpha }_2\hat{j}+{\alpha }_3\hat{k}$,

$\overline{b}=({\beta }_1,{\beta }_2,{\beta }_3)$

$\overline{c}=({\gamma }_1,{\gamma }_2,{\gamma }_3)$

then, $\left|\begin{array}{ccc}\overline{a}\cdot \overline{a}& \overline{a}\cdot \overline{b}& \overline{a}\cdot \overline{c}\\ \overline{b}\cdot \overline{a}& \overline{b}\cdot \overline{b}& \overline{b}\cdot \overline{c}\\ \overline{c}\cdot \overline{a}& \overline{c}\cdot \overline{b}& \overline{c}\cdot \overline{c}\end{array}\right|$

$=\left|\begin{array}{ccc}{\alpha }_1^2+{\alpha }_2^2+{\alpha }_3^2& {\alpha }_1{\beta }_1+{\alpha }_2{\beta }_2+{\alpha }_3{\beta }_3& {\alpha }_1{\gamma }_1+{\alpha }_2{\gamma }_2+{\alpha }_3{\gamma }_3\\ {\alpha }_1{\beta }_1+{\alpha }_2{\beta }_2+{\alpha }_3{\beta }_3& {\beta }_1^2+{\beta }_2^2+{\beta }_3^2& {\beta }_1{\gamma }_1+{\beta }_2{\gamma }_2+{\beta }_3{\gamma }_3\\ {\alpha }_1{\gamma }_1+{\alpha }_2{\gamma }_2+{\alpha }_3{\gamma }_3& {\beta }_1{\gamma }_1+{\beta }_2{\gamma }_2+{\beta }_3{\gamma }_3& {\gamma }_1^2+{\gamma }_2^2+{\gamma }_3^2\end{array}\right|$

$=\left|\begin{array}{ccc}{\alpha }_1& {\alpha }_2& \alpha 3\\ {\beta }_1& \beta 2& {\beta }_3\\ {\gamma }_1& {\gamma }_2& {\gamma }_3\end{array}\right|$$\left|\begin{array}{ccc}{\alpha }_1& {\beta }_1& {\gamma }_1\\ {\alpha }_2& {\beta }_2& {\gamma }_2\\ {\alpha }_3& {\beta }_3& {\gamma }_3\end{array}\right|$

$={\left[\overline{a}\overline{b}\overline{c}\right]}^2$