Answer
(a) $\langle u, v\rangle =1$.
(b) $d(u,v)=\sqrt{\frac{268}{9}}$.
Work Step by Step
$u=(0,3,\frac{1}{3}), \quad v=(\frac{4}{3},1,-3)$, $\langle u, v \rangle =2u_1v_1+ u_2v_2+2u_3v_3 $.
(a) $\langle u, v\rangle =2u_1v_1+ u_2v_2+2u_3v_3=0+3-2=1$.
(b) $d(u,v)=\| u-v \|=\| (-\frac{4}{3},2,\frac{10}{3}) \|=\sqrt{\frac{16\times 2}{9}+4+\frac{100\times 2 }{9}}=\sqrt{\frac{268}{9}}$.
