Answer
See the explanation below.
Work Step by Step
$u=(2,-\frac{1}{2},1), \quad v=(\frac{3}{2},2,-1)$, $\langle u, v \rangle =u_1v_1+2u_2v_2+3u_3v_3 $.
$$ \langle u, v\rangle =u_1v_1+2u_2v_2+3u_3v_3=3-2-3=-2$$ $$\| u \|=\sqrt{\langle u, u\rangle} =\sqrt{u^2_1+2u^2_2+3u^2_3}=\sqrt{4+\frac{1}{2}+3}=\sqrt{\frac{15}{2}}$$ $$\|v \|=\sqrt{\langle v, v\rangle} =\sqrt{v^2_1+2v^2_2+3v^2_3}=\sqrt{\frac{9}{4}+8+3}=\sqrt{\frac{53}{3}}$$
Now, we get
(a) $|\langle u, v \rangle|=2\leq\| u \|\| v \|=\sqrt{\frac{15}{2}}\sqrt{\frac{53}{3}} $
(b) $\| u+v \| =\| (\frac{7}{2},\frac{3}{2},0) \| =\sqrt{\frac{49}{4}+\frac{9\times 2}{4}}=\sqrt {\frac{67}{4}}\leq\| u \|+\| v \|=\sqrt{\frac{15}{2}}+\sqrt{\frac{53}{3}}.$
