Math Econ

Mathematic Economics

Test 07

one question

2015 Test 04

One questoin, prove KKT by using Farka’s lemma.

Old notes

Practice

Dot Product and Angle

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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