\(\displaystyle \left|r \vec{a}+s \vec{b}+ \vec{c}\right|^2 = r^2 \left|\vec{a}\right|^2+ s^2 \left|\vec{b}\right|^2 + \left|\vec{c}\right|^2 + 2rs\, \vec{a}\cdot \vec{b} + 2r\, \vec{a}\cdot \vec{c} + 2s\, \vec{b} \cdot \vec{c}\)
\(\displaystyle\qquad\qquad\qquad = 3 r^2 + 5 s^2 + 33 + 2rs +2r -18 s\)
\(\displaystyle\qquad\qquad\qquad = 3 r^2 + 2(s + 1)r + 5s^2-18s+33\)
\(\displaystyle\qquad\qquad\qquad =3(r+\frac{s+1}{3})^2 + 5s^2 - 18s+33 - \frac{s^2+2s+1}{3}\)
\(\displaystyle\qquad\qquad\qquad =3(r+\frac{s+1}{3})^2 + \frac{14}{3}( s^2 -4s + 7 )\)
\(\displaystyle\qquad\qquad\qquad =3(r+\frac{s+1}{3})^2 + \frac{14}{3}(s-2)^2 + 14\geq 14\)
且當 \(``=''\) 成立時， \(\displaystyle r+\frac{s+1}{3} = s-2=0\)，
亦即，當 \(s=2,\; r=-1\) 時，\(\displaystyle \left|r \vec{a}+s \vec{b}+ \vec{c}\right|\) 有最小值 \(\sqrt{14}.\)
