Answer
(a) $\| u \| = \sqrt{6}$
(b) $\| v \| = \sqrt{14}$
(c) $\langle u, v \rangle =1$
(d) $d(u,v)=\| u-v \|= \sqrt{18}.$
Work Step by Step
Let $u=( 1,-1,2), \quad v=( 2,3,1)$, then we have
(a) $\| u \| =\sqrt{\langle u, u\rangle}=\sqrt{u_1^2+u_2^2+u_3^2}=\sqrt{6}$
(b) $\| v \| =\sqrt{\langle v, v\rangle}=\sqrt{v_1^2+v_2^2+v_3^2}=\sqrt{14}$
(c) $\langle u, v \rangle =u_1v_1+u_2v_2+u_3v_3=1$
(d) $d(u,v)=\| u-v \|=\| (-1,-4,1) \|=\sqrt{18}.$
