Answer
(a) $\langle u, v \rangle = =0$
(b) $\| u \| = \sqrt{192}$
(c) $\| v \| = \sqrt{411}$
(d) $d(u,v)= \sqrt{603}$
Work Step by Step
$u=(8,0,-8), \quad v=(8,3,16)$
(a) $\langle u, v \rangle =2u_1v_1+3u_2v_2+u_3v_3=128+0-128=0$
(b) $\| u \| =\sqrt{\langle u, u\rangle}=\sqrt{2u_1^2+3u_2^2+u_3^2}=\sqrt{128+0+64}=\sqrt{192}$
(c) $\| v \| =\sqrt{\langle v, v\rangle}=\sqrt{2v_1^2+3v_2^2+v_3^2}=\sqrt{128+27+256}=\sqrt{411}$
(d) $d(u,v)=\| u-v \|=\| (0,-3,-24) \|=\sqrt{0+27+576 }=\sqrt{603}$
