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Dot product: invariance under rotation and equivalence of definitions

In this video I prove that the dot (scalar) product is invariant under rotation, then use this fact to show that the two commonly quoted definitions of the dot product (in terms of components and in terms of magnitudes & angles) are indeed equivalent. About me: I studied Physics at the University of Cambridge, then stayed on to get a PhD in Astronomy. During my PhD, I also spent four years teaching Physics undergraduates at the university. Now, I'm working as a private tutor, teaching Physics & Maths up to A Level standard. My website: https://benyelverton.com/ #mathematics #vectors #scalarproduct #scalar #product #dotproduct #rotation #invariant #basis #angle #magnitude #tensors #suffixnotation #summationconvention #orthogonal #matrix #trigonometry #geometry #alevel #furthermaths #maths #math #physics #science #education