 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2021-resit/9/iii/a/solution

Expand the dot product and cross product in Cartesian components:

u \cdot (v \times w) = u_{1} v_{2} w_{3} - u_{1} v_{3} w_{2} + u_{2} v_{3} w_{1} - u_{2} v_{1} w_{3} + u_{3} v_{1} w_{2} - u_{3} v_{2} w_{1} = (u_{2} v_{3} - u_{3} v_{2}) w_{1} + (u_{3} v_{1} - u_{1} v_{3}) w_{2} + (u_{1} v_{2} - u_{2} v_{1}) w_{3} = (u \times v) \cdot w .

The two expressions are the same scalar triple product, or determinant with columns

u, v, w

u \cdot (v \times w) = (u \times v) \cdot w .

The equality also holds when this determinant is zero.

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