Question Number 194218 by Mingma last updated on 30/Jun/23
Commented by Frix last updated on 30/Jun/23
$${t}=−\frac{\mathrm{3}}{\mathrm{2}} \\ $$
Commented by Mingma last updated on 30/Jun/23
Good!
Answered by cortano12 last updated on 01/Jul/23
$$\:\:\Rightarrow\overset{\rightarrow} {{a}}\:+\mathrm{t}\overset{\rightarrow} {{b}}\:=\:\begin{pmatrix}{\mathrm{1}+\mathrm{t}}\\{\mathrm{2}+\mathrm{t}}\\{\mathrm{3}+\mathrm{2t}}\end{pmatrix}\: \\ $$$$\:\Rightarrow\mid\overset{\rightarrow} {{a}}+\:\mathrm{t}\overset{\rightarrow} {{b}}\:\mid\:=\:\sqrt{\left(\mathrm{1}+\mathrm{t}\right)^{\mathrm{2}} +\left(\mathrm{2}+\mathrm{t}\right)^{\mathrm{2}} +\left(\mathrm{3}+\mathrm{2t}\right)^{\mathrm{2}} } \\ $$$$\:\:\mathrm{let}\:\mathrm{f}\left(\mathrm{t}\right)=\:\mathrm{t}^{\mathrm{2}} +\mathrm{2t}+\mathrm{1}+\mathrm{t}^{\mathrm{2}} +\mathrm{4t}+\mathrm{4}+\mathrm{4t}^{\mathrm{2}} +\mathrm{12t}+\mathrm{9} \\ $$$$\:\:\mathrm{f}\left(\mathrm{t}\right)\:=\:\mathrm{6t}^{\mathrm{2}} +\mathrm{18t}+\mathrm{14} \\ $$$$\:\:\mathrm{f}\left(\mathrm{t}\right)_{\mathrm{min}} \:\mathrm{when}\:\mathrm{t}=−\frac{\mathrm{18}}{\mathrm{12}}\:=\:−\frac{\mathrm{3}}{\mathrm{2}} \\ $$$$\:\:\underline{\:} \\ $$