Answer
(a) $|\langle u, v \rangle|=1\leq\| u \|\| v \|=\sqrt 17\sqrt 42 $
(b) $\| u+v \| =\sqrt{57}=7.55\leq\| u \|+\| v \|=\sqrt {17}+\sqrt {42}=4.12+6.48=10.6$.
Work Step by Step
Let $u=(1,0,4), \quad v=(-5,4,1)$, $\langle u, v \rangle=u\cdot v$, then we have
$\langle u, v \rangle =u_1v_1+u_1v_2+u_3v_3=-5+0+4=-1$,
$\| u \| =\sqrt{\langle u, u\rangle}=\sqrt{u_1^2+u_2^2+u_3^2}=\sqrt{1+0+16}=\sqrt{17}$,
$\| v \| =\sqrt{\langle v, v\rangle}=\sqrt{v_1^2+v_2^2+v_3^2}=\sqrt{25+16+1}=\sqrt{42}$,
$\| u+v \| =\| (-4,4,5) \| =\sqrt{16+16+25}=\sqrt{57}$
Now, we get
(a) $|\langle u, v \rangle|=1\leq\| u \|\| v \|=\sqrt 17\sqrt 42 $
(b) $\| u+v \| =\sqrt{57}=7.55\leq\| u \|+\| v \|=\sqrt {17}+\sqrt {42}=4.12+6.48=10.$6.
