Cho ba vectơ \(\overrightarrow{m}=\left(-5;4;9\right),\overrightarrow{n}=\left(2;-7;0\right),\overrightarrow{p}=\left(6;3;-4\right)\).
a) Tính \(\overrightarrow{m}.\overrightarrow{n},\overrightarrow{m}.\overrightarrow{p}\).
b) Tính \(\left|\overrightarrow{m}\right|,\left|\overrightarrow{n}\right|,\cos\left(\overrightarrow{m},\overrightarrow{n}\right)\).
c) Cho \(\overrightarrow{q}=\left(1;-2;0\right)\). Vectơ \(\overrightarrow{q}\) có vuông góc với \(\overrightarrow{p}\) không?
a) \(\overrightarrow m .\overrightarrow n = - 5.2 + 4.( - 7) = - 38\)
\(\overrightarrow m .\overrightarrow p = ( - 5).6 + 4.3 + 9.( - 4) = - 54\)
b) \(|\overrightarrow m | = \sqrt {{{( - 5)}^2} + {4^2} + {9^2}} = \sqrt {122} \)
\(|\overrightarrow n | = \sqrt {{2^2} + {{( - 7)}^2}} = \sqrt {53} \)
\(\cos (\overrightarrow m ,\overrightarrow n ) = \frac{{\overrightarrow m .\overrightarrow n }}{{|\overrightarrow m |.|\overrightarrow n |}} = \frac{{ - 38}}{{\sqrt {122} .\sqrt {53} }} = - \frac{{19\sqrt {6466} }}{{3233}}\)
c) \(\overrightarrow q .\overrightarrow p = 1.6 + 3.(-2) - 4.0 = 0\) nên \(\overrightarrow q \) vuông góc với \(\overrightarrow p \).