 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2013/paper-2/12/a/i/solution

Using the Einstein summation convention, reorder the scalar components and sum over the two dummy indices separately:

a_{i} b_{k} b_{i} a_{k} = (a_{i} b_{i}) (a_{k} b_{k}) = (a \cdot b)^{2} .

The repeated suffixes each represent the three-dimensional component sum.

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