Question

show that the vector i -2j +3k, -2i +3j -4k, i -3j +5k are coplanar​

Answer 1

$\text{Let}$

$\overrightarrow{a}=\overrightarrow{i} -2\overrightarrow{j}+3\overrightarrow{k}$

$\overrightarrow{b}=-2\overrightarrow{i}+3\overrightarrow{j}-4\overrightarrow{k}$

$\overrightarrow{c}=\overrightarrow{i} -3\overrightarrow{j}+5\overrightarrow{k}$

$[\overrightarrow{a}\;\overrightarrow{b}\;\overrightarrow{c}]$

$=\left|\begin{array}{ccc}1&-2&3\\-2&3&-4\\1&-3&5\end{array}\right|$

$=1(15-12)+2(-10+4)+3(6-3)$

$=3-12+9$

$=0$

$\therefore\overrightarrow{a},\overrightarrow{b}\;\text{and}\,\overrightarrow{c}\;\text{are coplanar}$