Question

If$\overrightarrow{\mathbf{u}}\mathbf{,}\overrightarrow{\mathbf{v}}\mathbf{,}\overrightarrow{\mathbf{w}}$are three unit vectors in${\mathbf{ℝ}}^{\mathbf{3}}$, what are the possible values of$\mathit{d}\mathit{e}\mathit{t}\left[\begin{array}{ccc}\overrightarrow{u}& \overrightarrow{v}& \overrightarrow{w}\end{array}\right]$?

Short answer

Therefore, the possible values of $det\left[\begin{array}{ccc}\overrightarrow{\mathrm{u}}& \overrightarrow{\mathrm{v}}& \overrightarrow{\mathrm{w}}\end{array}\right]$is given by,

$det\left[\begin{array}{ccc}\overrightarrow{\mathrm{u}}& \overrightarrow{\mathrm{v}}& \overrightarrow{\mathrm{w}}\end{array}\right]\in \left[-1,1\right]$

Step-by-step solution

Given. 

Consider, the given unit vector,

Unit Vectors $=\overrightarrow{u},\overrightarrow{v},\overrightarrow{w}$.

To find the possible value of det[u→v→w→].

If$\overrightarrow{u},\overrightarrow{v},\overrightarrow{w}$ are unit vectors, then

$det\left[\begin{array}{ccc}\overrightarrow{u}& \overrightarrow{v}& \overrightarrow{w}\end{array}\right]=\left(\overrightarrow{u}\times \overrightarrow{v}\times \overrightarrow{w}\right)$

Since $\overrightarrow{u}$and $\overrightarrow{v}$are unit vectors, then $\overrightarrow{u}\times \overrightarrow{v}$is also a unit vector. If $\phi$is the angle between $\overrightarrow{u}\times \overrightarrow{v}$and $\overrightarrow{w}$, then

$\cos \phi =\frac{\left(\overrightarrow{u}\times \overrightarrow{v}\times \overrightarrow{w}\right)}{||\overrightarrow{u}\times \overrightarrow{v}||||\overrightarrow{w}||}=\left(\overrightarrow{u}\times \overrightarrow{v}\times \overrightarrow{w}\right)=det\left[\begin{array}{ccc}\overrightarrow{u}& \overrightarrow{v}& \overrightarrow{w}\end{array}\right]$, which means that the right side of this equation can vary between -1 and 1.