EXERCISES

1. (10) Explain the geometrical significance for the vectors $x$ and $y$ of:
   1. 
      
      $$x \cdot y = 0.$$
      
      

   2. 
      
      $$x \cdot y \leq 0.$$
      
      

   3. 
      
      $$\cos \theta = \frac {x \cdot y}{\Vert x\Vert\Vert y\Vert},$$
      
      
      where $\theta$ is an angle.
2. (10) Prove that the function $f: \Re^n \rightarrow \Re$ defined by $f(x) = a \cdot x+\alpha$ is continuous, where $a$ is a given vector and $\alpha \in \Re$.
3. (10) Prove that the function $f: \Re^n \rightarrow \Re$ defined by $f(x) = \Vert x\Vert$ is continuous.