Math

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Dot Products, Norms, and Angles Between Vectors.

Recall that the Law of Cosines, a generalization on the Pythagorean Theorem, gives us the relationship between the side lengths of an arbitrary triangle. Specifically, if a triangle has side lengths aa, bb, and cc, then

$$a^2 + b^2 - 2ab\cos \theta = c^2$$

where θ is the angle between the sides of length aa and bb.

Consider the triangle that can be formed from the vectors xx, yy, and x−yx−y.

Applying the Law of Cosines to this triangle, we have $||x||^2 +||y||^2 - 2||x|| \, ||y||\cos \theta = ||x-y||^2$

But this implies, using our observations about the dot product made above, that

$$\begin{array}{rcl} (x \cdot x) + (y \cdot y) - 2||x|| \, ||y||\cos \theta & = & (x-y) \cdot (x-y)\\ & = & x \cdot (x-y) - y \cdot (x-y)\\ & = & (x \cdot x) - (x \cdot y) - (y \cdot x) + (y \cdot y)\\ & = & (x \cdot x) - (x \cdot y) - (x \cdot y) + (y \cdot y)\\ & = & (x \cdot x) - 2(x \cdot y) + (y \cdot y)\\ \end{array}$$

Subtracting the common (x⋅x)(x⋅x) and (y⋅y)(y⋅y) from both sides, we find −2||x||||y||cosθ=−2(x⋅y)

Which, solving for cosθcos⁡θ tells us

$$\cos \theta = \frac{x \cdot y}{||x|| \, ||y||}$$

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