Question

Let $\overrightarrow{a_r}=x_r\hat{i}+y_r\hat{j}+z_r\hat{k}$, $r=1,2,3$ be mutually perpendicular unit vectors, then the value of $\left|\begin{array}{ccc}{x}_1& x_2& x_3\\ y_1& y_2& y_3\\ z_1& z_2& z_3\end{array}\right|$ is equal to

• A. zero
• B. $\pm 1$
• C. $\pm 2$
• D. none of these

Answer 1

Answer: (B)

$\pm 1$

$\overrightarrow{a_r}=x_r\hat{i}+y_r\hat{j}+z_r\hat{k}$, $r=1,2,3$ be mutually perpendicular unit vectors.
i.e $x_r^2+y_r^2+z_r^2=1$ (since given vectors are unit vectors)
$x_1{x}_2+y_1{y}_2+z_1{z}_2=\mathrm{0...}etc$ (since vectors are mutually perpendicular)

$\left|\begin{array}{ccc}{x}_1& x_2& x_3\\ y_1& y_2& y_3\\ z_1& z_2& z_3\end{array}\right|\times \left|\begin{array}{ccc}{x}_1& x_2& x_3\\ y_1& y_2& y_3\\ z_1& z_2& z_3\end{array}\right|$ $=\left|\begin{array}{ccc}{x}_1& y_1& z_1\\ x_2& y_2& z_2\\ x_3& y_3& z_3\end{array}\right|\times \left|\begin{array}{ccc}{x}_1& x_2& x_3\\ y_1& y_2& y_3\\ z_1& z_2& z_3\end{array}\right|$ $=\left|\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right|=1$

$\Rightarrow {\left|\begin{array}{ccc}{x}_1& x_2& x_3\\ y_1& y_2& y_3\\ z_1& z_2& z_3\end{array}\right|}^2=1$

$\therefore \left|\begin{array}{ccc}{x}_1& x_2& x_3\\ y_1& y_2& y_3\\ z_1& z_2& z_3\end{array}\right|=\pm 1$