Bài 1:
a: \(\overrightarrow{a}=2\overrightarrow{i}-\overrightarrow{j};\overrightarrow{b}=\overrightarrow{i}+4\overrightarrow{j};c=-\overrightarrow{i}-2\overrightarrow{j}\)
=>\(\overrightarrow{a}=\left(2;-1\right);\overrightarrow{b}=\left(1;4\right);\overrightarrow{c}=\left(-1;-2\right)\)
\(\overrightarrow{u}=2\cdot\overrightarrow{a}+3\cdot\overrightarrow{b}-\overrightarrow{c}\)
=>\(\overrightarrow{u}=\left(2\cdot2+3\cdot1-\left(-1\right);2\cdot\left(-1\right)+3\cdot4-\left(-2\right)\right)\)
=>\(\overrightarrow{u}=\left(8;12\right)\)
\(\left|\overrightarrow{u}\right|=\sqrt{8^2+12^2}=\sqrt{144+64}=\sqrt{208}=4\sqrt{13}\)
b: Đặt \(\overrightarrow{c}=x\cdot\overrightarrow{a}+y\cdot\overrightarrow{b}\)
\(\overrightarrow{a}=\left(2;-1\right);\overrightarrow{b}=\left(1;4\right);\overrightarrow{c}=\left(-1;-2\right)\)
=>\(\left\{{}\begin{matrix}-1=x\cdot2+y\cdot1\\-2=x\cdot\left(-1\right)+4y\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}2x+y=-1\\-x+4y=-2\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}2x+y=-1\\-2x+8y=-4\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}9y=-5\\x-4y=2\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}y=-\dfrac{5}{9}\\x=4y+2=4\cdot\left(-\dfrac{5}{9}\right)+2=-\dfrac{20}{9}+2=-\dfrac{2}{9}\end{matrix}\right.\)
vậy: \(\overrightarrow{c}=\dfrac{-2}{9}\cdot\overrightarrow{a}+\dfrac{-5}{9}\cdot\overrightarrow{b}\)
