Question

Question:A basis$\overrightarrow{{\mathbf{v}}_{\mathbf{1}}}\mathbf{,}\overrightarrow{{\mathbf{v}}_{\mathbf{2}}}\mathbf{,}\overrightarrow{{\mathbf{v}}_{\mathbf{3}}}$of${\mathbf{ℝ}}^{\mathbf{3}}$is called positively oriented if$\overrightarrow{{\mathbf{v}}_{\mathbf{1}}}$encloses an acute angle with$\overrightarrow{{\mathbf{v}}_{\mathbf{2}}}\mathbf{\times }\overrightarrow{{\mathbf{v}}_{\mathbf{3}}}$. Illustrate this definition with a sketch. Show that the basis is positively oriented if (and only if)$\mathit{d}\mathit{e}\mathit{t}\left[\overrightarrow{v_1},\overrightarrow{v_2},\overrightarrow{v_3}\right]$is positive.

Short answer

Thus, this basis is positively oriented if and only if$\det [\overrightarrow{{\mathrm{v}}_1},\overrightarrow{{\mathrm{v}}_2},\overrightarrow{{\mathrm{v}}_3}]$>0

Step-by-step solution

Definition.

A determinant is a unique number associatedwith a square matrix.

A determinant is a scalar value that is a function of the entries of a square matrix.

It is the signed factor by which areas are scaled by this matrix. If the sign is negative the matrix reverses orientation.

Show that [v1→, v2→, v3→] is positive.

A basis $\left\{\overrightarrow{{\mathrm{v}}_1},\overrightarrow{{\mathrm{v}}_2},\overrightarrow{{\mathrm{v}}_3}\right\}$for ${\mathrm{ℝ}}^{3^{}}$is positively oriented if$\overrightarrow{{\mathrm{v}}_1}$ encloses an acute angle with $\overrightarrow{{\mathrm{v}}_2}\times \overrightarrow{{\mathrm{v}}_3}$.

We know that

$\det [\overrightarrow{{\mathrm{v}}_1},\overrightarrow{{\mathrm{v}}_2},\overrightarrow{{\mathrm{v}}_3}]=\overrightarrow{{\mathrm{v}}_1}·\left(\overrightarrow{{\mathrm{v}}_2},\overrightarrow{{\mathrm{v}}_3}\right),$

which is positive if and only if the angle between $\overrightarrow{{\mathrm{v}}_1}\mathrm{and}\overrightarrow{{\mathrm{v}}_2},\overrightarrow{{\mathrm{v}}_3}$is acute.

Thus, this basis is positively oriented if and only if$\det [\overrightarrow{{\mathrm{v}}_1}\overrightarrow{{\mathrm{v}}_2}\overrightarrow{{\mathrm{v}}_3}]>0$