 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2017/prelims/mathematics-iv-resit/1/a/solution

The scalar product, also called the dot product, and the vector product, also called the cross product, are

a \cdot b = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3}, a \land b = a_{2} b_{3} - a_{3} b_{2} a_{3} b_{1} - a_{1} b_{3} a_{1} b_{2} - a_{2} b_{1} .

For nonzero vectors, let

ϑ \in [0, π]

be their angle. Their Euclidean norms satisfy

a \cdot b = ∣ a ∣∣ b ∣ cos ϑ, ∣ a \land b ∣ = ∣ a ∣∣ b ∣ sin ϑ .

The cross product is orthogonal to both vectors, with its orientation determined by the right-hand rule. If either vector is zero, both products vanish, although the angle is undefined.

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