cho 2 vecto \(\overrightarrow{a}=\left(1;7\right);\overrightarrow{b}=\left(7;-1\right)\) và \(\overrightarrow{c}=\left(4;m-4\right)\)
a) Tính \(\left|\overrightarrow{a}\right|\)
b) Tính \(\left|\overrightarrow{b}\right|\)
c) Tính \(\overrightarrow{a}.\overrightarrow{b}\)
d) chứng minh \(\overrightarrow{a}\perp\overrightarrow{b}\)
e) tìm m để \(\overrightarrow{a}\perp\overrightarrow{c}\)
f) tìm m để \(\overrightarrow{b}\) cùng phương với \(\overrightarrow{c}\)
g) tìm m để \(\left|\overrightarrow{c}\right|=5\)
a: \(\left|\overrightarrow{a}\right|=\sqrt{1^2+7^2}=\sqrt{50}=5\sqrt{2}\)
b: \(\left|\overrightarrow{b}\right|=\sqrt{7^2+\left(-1\right)^2}=\sqrt{50}=5\sqrt{2}\)
c: \(\overrightarrow{a}\cdot\overrightarrow{b}=1\cdot7+7\cdot\left(-1\right)=7-7=0\)
d: \(\overrightarrow{a}\cdot\overrightarrow{b}=1\cdot7+7\cdot\left(-1\right)=7-7=0\)
=>\(\overrightarrow{a}\perp\overrightarrow{b}\)
e: \(\overrightarrow{a}\perp\overrightarrow{c}\)
=>\(1\cdot4+7\cdot\left(m-4\right)=0\)
=>7(m-4)+4=0
=>7m-28+4=0
=>7m-24=0
=>7m=24
=>\(m=\dfrac{24}{7}\)
f: \(\overrightarrow{b};\overrightarrow{c}\) cùng phương khi \(\dfrac{7}{4}=\dfrac{-1}{m-4}\)
=>\(m-4=-\dfrac{4}{7}\)
=>\(m=4-\dfrac{4}{7}=\dfrac{24}{7}\)
g: \(\left|\overrightarrow{c}\right|=5\)
=>\(\sqrt{4^2+\left(m-4\right)^2}=5\)
=>\(\left(m-4\right)^2+16=25\)
=>\(\left(m-4\right)^2=9\)
=>\(\left[{}\begin{matrix}m-4=3\\m-4=-3\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}m=7\\m=1\end{matrix}\right.\)