Algebra and Geometry | Part IA, 2005

Convert the following expressions from suffix notation (assuming the summation convention in three dimensions) into standard notation using vectors and/or matrices, where possible, identifying the one expression that is incorrectly formed:

(i) δij\delta_{i j},

(ii) δiiδij\delta_{i i} \delta_{i j},

(iii) δllaibjCijdkCikdi\delta_{l l} a_{i} b_{j} C_{i j} d_{k}-C_{i k} d_{i},

(iv) ϵijkakbj\epsilon_{i j k} a_{k} b_{j},

(v) ϵijkajak\epsilon_{i j k} a_{j} a_{k}.

Write the vector triple product a×(b×c)\mathbf{a} \times(\mathbf{b} \times \mathbf{c}) in suffix notation and derive an equivalent expression that utilises scalar products. Express the result both in suffix notation and in standard vector notation. Hence or otherwise determine a×(b×c)\mathbf{a} \times(\mathbf{b} \times \mathbf{c}) when a\mathbf{a} and b\mathbf{b} are orthogonal and c=a+b+a×b\mathbf{c}=\mathbf{a}+\mathbf{b}+\mathbf{a} \times \mathbf{b}.

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