Algebra and Geometry | Part IA, 2003

Derive an expression for the triple scalar product (e1×e2)e3\left(\mathbf{e}_{1} \times \mathbf{e}_{2}\right) \cdot \mathbf{e}_{3} in terms of the determinant of the matrix EE whose rows are given by the components of the three vectors e1,e2,e3\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}.

Use the geometrical interpretation of the cross product to show that ea,a=1,2,3\mathbf{e}_{a}, a=1,2,3, will be a not necessarily orthogonal basis for R3\mathbb{R}^{3} as long as detE0\operatorname{det} E \neq 0.

The rows of another matrix E^\hat{E} are given by the components of three other vectors e^b,b=1,2,3\hat{\mathbf{e}}_{b}, b=1,2,3. By considering the matrix EE^TE \hat{E}^{\mathrm{T}}, where T{ }^{\mathrm{T}} denotes the transpose, show that there is a unique choice of E^\hat{E} such that e^b\hat{\mathbf{e}}_{b} is also a basis and

eae^b=δab\mathbf{e}_{a} \cdot \hat{\mathbf{e}}_{b}=\delta_{a b}

Show that the new basis is given by

e^1=e2×e3(e1×e2)e3 etc. \hat{\mathbf{e}}_{1}=\frac{\mathbf{e}_{2} \times \mathbf{e}_{3}}{\left(\mathbf{e}_{1} \times \mathbf{e}_{2}\right) \cdot \mathbf{e}_{3}} \quad \text { etc. }

Show that if either ea\mathbf{e}_{a} or e^b\hat{\mathbf{e}}_{b} is an orthonormal basis then EE is a rotation matrix.

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