Mathematical Immaturity

1.13 Exercises

Prove that each of the statements in Exercises 3 through 7 is valid for all elements $x$ and $y$ in a real Euclidean space.

• If $x$ and $y$ are nonzero elements making an angle $\theta$ with each other, then $$\|x - y\|^2 = \|x\|^2 + \|y\|^2 - 2 \|x\| \|y\| \cos \theta.$$
• Recall from Section 11:
  $1. \quad$For an element $x$ in a Euclidean space $V$ with inner product $(x, x),$ the nonnegative number $\|x\| = (x, x)^{1/2}$ is called the norm of $x.$
  
  $2.\quad$ In a real Euclidean space $V,$ the angle between two nonzero elements $x$ and $y$ is defined to be that number $\theta$ in the interval $0 \leq \theta \leq \pi$ which satisfies the equation \begin{align*} \cos \theta &= \frac{(x, y)}{\|x\|\|y\|}. \end{align*}
• 
  Proof.$\quad$We have \begin{align*} \|x - y\|^2 &= (x - y, x - y) \\ &= (x, x) - 2(x, y) + (y, y) \\ &= \|x\|^2 + \|y\|^2 - 2\|x\|\|y\|\frac{(x, y)}{\|x\|\|y\|} \\ &= \|x\|^2 + \|y\|^2 - 2\|x\|\|y\|\cos \theta \quad \blacksquare \end{align*}