## 311 – HW5

(I am counting as HW3 the homework exercises supervised by Sam Coskey during the two weeks following Isabel’s birth, and as HW4 the two written exercises assigned by Sam that I collected on March 17.)

This exercise is due April 7 at the beginning of lecture.

Provide a proof verifying that the function $d:\mathbb R^2\times\mathbb R^2\to\mathbb R$ given by

$d((x,y),(z,w))=\sqrt{2(x-z)^2+2(y-w)^2-3(x-z)(y-w)}$

is a distance function.

### One Response to 311 – HW5

1. To clarify: Points in $\mathbb R^2$ are pairs $(a,b)$ where $a,b\in\mathbb R$. For instance, to verify the triangle inequality one needs to prove that if $A,B,C$ are three points in $\mathbb R^2$, then $d(A,C)\le d(A,B)+d(B,C)$. To do this, one needs to assign coordinates to $A,B,C$, say $A=(x,y),B=(z,w),C=(u,v)$, and proceed algebraically using the definition of $d$ given in the post.

(The high level reason why the function $d$ defines a distance is that it comes from a norm on $\mathbb R^2$ given by the quadratic form of a positive definite matrix, but rather than appealing to such machinery, I expect direct algebraic verifications.)