Câu 1 :
\(\overrightarrow{u}\left(2m-1;-2\right)\uparrow\uparrow\overrightarrow{v}=\left(-2;m+3\right)\)
\(\Leftrightarrow\dfrac{2m-1}{-2}=\dfrac{-2}{m+3}\)
\(\Leftrightarrow2m^2+5m-3=4\Leftrightarrow2m^2+5m-7=0\Leftrightarrow\left[{}\begin{matrix}m=1\\m=-\dfrac{7}{2}\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}\overrightarrow{u}=\left(1;-2\right)\\\overrightarrow{u}=\left(-8;-2\right)\end{matrix}\right.\)
Câu 6 : \(E\left(x;y\right)\)
\(\overrightarrow{AE}=-2\overrightarrow{AB}+3\overrightarrow{AC}\)
\(\Leftrightarrow\left(x-2;y-2\right)=-2\left(-1;-5\right)+3\left(-5;-2\right)\)
\(\Leftrightarrow\left\{{}\begin{matrix}x-2=2-15\\y-2=10-6\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x=-11\\y=6\end{matrix}\right.\) \(\Rightarrow E\left(-11;6\right)\)
Câu 15 :
\(\overrightarrow{u}=2\overrightarrow{a}-\overrightarrow{b}\Leftrightarrow\overrightarrow{u}=\left(2.1-5;2.2-3\right)=\left(-3;-1\right)\)
\(\overrightarrow{v}=\overrightarrow{a}+3\overrightarrow{b}\Leftrightarrow\overrightarrow{v}=\left(1+3.5;2+3.3\right)=\left(16;11\right)\)
Câu 1: \(\overrightarrow{u};\overrightarrow{v}\) cùng phương
=>\(\dfrac{2m-1}{-2}=\dfrac{-2}{m+3}\)
=>\(\left(2m-1\right)\left(m+3\right)=4\)
=>\(2m^2+6m-m-3-4=0\)
=>\(2m^2+5m-7=0\)
=>\(2m^2+7m-2m-7=0\)
=>(2m+7)(m-1)=0
=>\(\left[{}\begin{matrix}2m+7=0\\m-1=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}m=-\dfrac{7}{2}\\m=1\end{matrix}\right.\)
Câu 6:
A(2;3); B(1;-3); C(-3;0); E(x;y)
\(\overrightarrow{AE}=\left(x-2;y-3\right);\overrightarrow{AB}=\left(1-2;-3-3\right)=\left(-1;-6\right)\)
\(\overrightarrow{AC}=\left(-3-2;0-3\right)=\left(-5;-3\right)\)
\(\overrightarrow{AE}=-2\cdot\overrightarrow{AB}+3\cdot\overrightarrow{AC}\)
=>\(\left\{{}\begin{matrix}x-2=-2\cdot\left(-1\right)+3\cdot\left(-5\right)=2-15=-13\\y-3=-2\cdot\left(-6\right)+3\cdot\left(-3\right)=12-9=3\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}x=-13+2=-11\\y=3+3=6\end{matrix}\right.\)
=>E(-11;6)
Câu 15:
\(\overrightarrow{u}=2\overrightarrow{a}-\overrightarrow{b}\)
=>\(\left\{{}\begin{matrix}x_{\overrightarrow{u}}=2\cdot1-5=2-5=-3\\y_{\overrightarrow{u}}=2\cdot2-3=4-3=1\end{matrix}\right.\)
=>\(\overrightarrow{u}=\left(-3;1\right)\)
\(\overrightarrow{v}=\overrightarrow{a}+3\cdot\overrightarrow{b}\)
=>\(\left\{{}\begin{matrix}x_{\overrightarrow{v}}=1+3\cdot5=16\\y_{\overrightarrow{v}}=2+3\cdot3=2+9=11\end{matrix}\right.\)
=>\(\overrightarrow{v}=\left(16;11\right)\)
