Prove angle addition holds for Rn Define θ(u,v)=cos−1(u⋅v|u||v|) be the angle between u,v∈Rn, where u⋅v is the standard inner product and |x|=x⋅x for all x∈Rn. Let w=su+tv where s,t≥0 are scalars. Show θ(u,v)=θ(u,w)+θ(v,w) for u,v≠0

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Prove angle addition holds for Rn  Define θ(u,v)=cos−1(u⋅v|u||v|) be the angle between u,v∈Rn, where u⋅v is the standard inner product and |x|=x⋅x for all x∈Rn. Let w=su+tv where s,t≥0 are scalars. Show θ(u,v)=θ(u,w)+θ(v,w) for u,v≠0

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Assume |u|=|v|=1 and w=su+tv, and denote u⋅v=x. Then |w|=s2+t2+2stx,cos⁡θ(u,w)=s+txs2+t2+2stx and cos⁡θ(v,w)=t+sxs2+t2+2stx. Also sin⁡θ(u,w)=1−cos2θ(u,w)=s2+t2+2stx−s2−t2x2−2stxs2+t2+2stx=t1−x2s2+t2+2stx  and similarly sin⁡θ(v,w)=s1−x2s2+t2+2stx. Now  cos⁡(θ(u,w)+θ(v,w))=cos⁡θ(u,w)cos⁡θ(v,w)−sin⁡θ(u,w)sin⁡θ(v,w)=  =st+s2x+t2x+stx2−st(1−x2)s2+t2+2stx=(s2+t2+2stx)xs2+t2+2stx=x=cos⁡θ(u,v)

Prove angle addition holds for \mathbb{R}^n Define \theta(u,v)=\cos^{-1}(\frac{u \cdot v}{|u||v|}) be

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