Paper 1, Section II, 5C5 \mathrm{C}

Vectors and Matrices | Part IA, 2009

Let a,b,c\mathbf{a}, \mathbf{b}, \mathbf{c} be unit vectors. By using suffix notation, prove that

(a×b)(a×c)=bc(ab)(ac)(\mathbf{a} \times \mathbf{b}) \cdot(\mathbf{a} \times \mathbf{c})=\mathbf{b} \cdot \mathbf{c}-(\mathbf{a} \cdot \mathbf{b})(\mathbf{a} \cdot \mathbf{c})


(a×b)×(a×c)=[a(b×c)]a(\mathbf{a} \times \mathbf{b}) \times(\mathbf{a} \times \mathbf{c})=[\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})] \mathbf{a}

The three distinct points A,B,CA, B, C with position vectors a,b,c\mathbf{a}, \mathbf{b}, \mathbf{c} lie on the surface of the unit sphere centred on the origin OO. The spherical distance between the points AA and BB, denoted δ(A,B)\delta(A, B), is the length of the (shorter) arc of the circle with centre OO passing through AA and BB. Show that

cosδ(A,B)=ab\cos \delta(A, B)=\mathbf{a} \cdot \mathbf{b}

A spherical triangle with vertices A,B,CA, B, C is a region on the sphere bounded by the three circular arcs AB,BC,CAA B, B C, C A. The interior angles of a spherical triangle at the vertices A,B,CA, B, C are denoted α,β,γ\alpha, \beta, \gamma, respectively.

By considering the normals to the planes OABO A B and OACO A C, or otherwise, show that

cosα=(a×b)(a×c)a×ba×c\cos \alpha=\frac{(\mathbf{a} \times \mathbf{b}) \cdot(\mathbf{a} \times \mathbf{c})}{|\mathbf{a} \times \mathbf{b} \| \mathbf{a} \times \mathbf{c}|}

Using identities (1) and (2), prove that

cosδ(B,C)=cosδ(A,B)cosδ(A,C)+sinδ(A,B)sinδ(A,C)cosα\cos \delta(B, C)=\cos \delta(A, B) \cos \delta(A, C)+\sin \delta(A, B) \sin \delta(A, C) \cos \alpha


sinαsinδ(B,C)=sinβsinδ(A,C)=sinγsinδ(A,B)\frac{\sin \alpha}{\sin \delta(B, C)}=\frac{\sin \beta}{\sin \delta(A, C)}=\frac{\sin \gamma}{\sin \delta(A, B)}

For an equilateral spherical triangle show that α>π/3\alpha>\pi / 3.

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