Question Number 135423 by benjo_mathlover last updated on 13/Mar/21
$${If}\:\overset{\rightarrow} {{a}}=\left(\mathrm{4},\mathrm{2},−\mathrm{1}\right),\:\overset{\rightarrow} {{b}}=\left({m},\mathrm{1},\mathrm{1}\right) \\ $$$$\overset{\rightarrow} {{c}}=\left(\bar {\mathrm{3}}−\mathrm{1},\mathrm{0}\right)\:{are}\:{three}\:{vectors} \\ $$$${then}\:{find}\:{the}\:{value}\:{of}\:{m}\:{such} \\ $$$${that}\:\overset{\rightarrow} {{a}},\overset{\rightarrow} {{b}}\:{and}\:\overset{\rightarrow} {{c}}\:{are}\:{coplanar}\:{and} \\ $$$${find}\:\overset{\rightarrow} {{a}}×\left(\overset{\rightarrow} {{b}}×\overset{\rightarrow} {{c}}\right). \\ $$
Answered by mr W last updated on 13/Mar/21
$${a}×{c}=\begin{vmatrix}{\mathrm{4}}&{\mathrm{2}}&{−\mathrm{1}}\\{\mathrm{3}}&{−\mathrm{1}}&{\mathrm{0}}\end{vmatrix}=\left(−\mathrm{1},−\mathrm{3},−\mathrm{10}\right) \\ $$$$\left({a}×{c}\right)\centerdot{b}=−\mathrm{1}×{m}−\mathrm{3}×\mathrm{1}−\mathrm{10}×\mathrm{1}=\mathrm{0} \\ $$$$\Rightarrow{m}=−\mathrm{13} \\ $$$$… \\ $$