Question

Value of $\left|\begin{array}{ccc}\overline{a}.\overline{a}& \overline{a}.\overline{b}& \overline{a}.\overline{c}\\ \overline{b}.\overline{a}& \overline{b}.\overline{b}& \overline{b}.\overline{c}\\ \overline{c}.\overline{a}& \overline{c}.\overline{b}& \overline{c}.\overline{c}\end{array}\right|$ is

• A. $\left[\overline{a}\overline{b}\overline{c}\right]$
• B. ${\left[\overline{a}\overline{b}\overline{c}\right]}^2$
• C. ${\left[\overline{a}\overline{b}\overline{c}\right]}^3$
• D. $3\left[\overline{a}\overline{b}\overline{c}\right]$

Answer 1

Answer: (B)

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

Let $\overline{a}=x_1\hat{i}+y_1\hat{j}+z_1\hat{k},\overline{b}=x_2\hat{i}+y_2\hat{j}+z_2\hat{k}$ and $\overline{c}=x_3\hat{i}+y_3\hat{j}+z_3\hat{k}$

$\left|\begin{array}{ccc}\overline{a}.\overline{a}& \overline{a}.\overline{b}& \overline{a}.\overline{c}\\ \overline{b}.\overline{a}& \overline{b}.\overline{b}& \overline{b}.\overline{c}\\ \overline{c}.\overline{a}& \overline{c}.\overline{b}& \overline{c}.\overline{c}\end{array}\right|=\left|\begin{array}{ccc}{x}_1^2+y_1^2+z_1^2& x_1{x}_2+y_1{y}_2+z_1{z}_2& x_1{x}_3+y_1{y}_3+z_1{z}_3\\ x_1{x}_2+y_1{y}_2+z_1{z}_2& x_2^2+y_2^2+z_2^2& x_2{x}_3+y_2{y}_3+z_2{z}_3\\ x_1{x}_3+y_1{y}_3+z_1{z}_3& x_2{x}_3+y_2{y}_3+z_2{z}_3& x_3^2+y_3^2+z_3^2\end{array}\right|$

$=\left|\begin{array}{ccc}{x}_1& y_1& z_1\\ x_2& y_2& z_2\\ x_3& y_3& z_3\end{array}\right|\left|\begin{array}{ccc}{x}_1& y_1& z_1\\ x_2& y_2& z_2\\ x_3& y_3& z_3\end{array}\right|$

$\therefore \left|\begin{array}{ccc}\overline{a}.\overline{a}& \overline{a}.\overline{b}& \overline{a}.\overline{c}\\ \overline{b}.\overline{a}& \overline{b}.\overline{b}& \overline{b}.\overline{c}\\ \overline{c}.\overline{a}& \overline{c}.\overline{b}& \overline{c}.\overline{c}\end{array}\right|={\left[\overline{a}\overline{b}\overline{c}\right]}^2$

Hence, option B.