1. 

1. (a) 0°

2. (b) 90°

2. 

1. (a) $\sqrt{14}$ (scalar projection), ${(2,1,3)}^T$ (vector projection);

2. (b) 0, 0;

3. (c) $\frac{{14\sqrt{13}}^T}{13},{(\frac{42}{13},\frac{28}{13})}^T$;

4. $\frac{{8\sqrt{21}}^T}{21},{(\frac{8}{21},\frac{16}{21},\frac{32}{21})}^T$

3. 

1. (a)
   $\begin{array}{c}\mathbf{p}=(3,0){}^T,\mathbf{x}-\mathbf{p}={(0,4)}^T,\\ {\mathbf{p}}^T(\mathbf{x}-\mathbf{p})=3.0+0.4=0;\hfill \end{array}$

2. (c)
   $\begin{array}{c}\mathbf{p}=(3,3,3){}^T,\mathbf{x}-\mathbf{p}={(-1,1,0)}^T,\hfill \\ {\mathbf{p}}^T(\mathbf{x}-\mathbf{p})=-1\cdot 3+1\cdot 3+0\cdot 3=0\end{array}$

5. $(1.8,3.6)$

6. $(1.4,3.8)$

7. 0.4

8. 

1. (a) $2x+4y+3z=0$;

2. (c) $z-4=0$

9. $\frac{5}{3}$

10. $\frac{8}{7}$

13. Hint: Expand the right hand side of the equation

20. The correlation matrix with entries rounded to two decimal places is