# The lines $\overrightarrow r=\overrightarrow a_1+\lambda\overrightarrow b_1$ and $\overrightarrow r=\overrightarrow a_2+\alpha \overrightarrow b_2$ are coplanar if ?

$\begin{array}{1 1} (a)\:\overrightarrow a_1\times\overrightarrow a_2=0\:\qquad\:(b)\:\overrightarrow b_1\times\overrightarrow b_2=0\:\qquad\:(c)\:(\overrightarrow a_1-\overrightarrow a_2)\times(\overrightarrow b_1-\overrightarrow b_2)=0\:\qquad\:(d)\:[\overrightarrow a_1\:\overrightarrow b_1\:\overrightarrow b_2]=[\overrightarrow a_2\:\overrightarrow b_1\:\overrightarrow b_2] \end{array}$

Toolbox:
• $[\overrightarrow a\:\overrightarrow b\:\overrightarrow c]=\left| \begin {array} {ccc} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3\\ c_1 & c_2 & c_3\end {array} \right |$
• $\left| \begin {array} {ccc} a_1-d_1 & a_2-d_2 & a_3-d_3 \\ b_1 & b_2 & b_3\\ c_1 & c_2 & c_3\end {array} \right |=\left| \begin {array} {ccc} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3\\ c_1 & c_2 & c_3\end {array} \right |-\left| \begin {array} {ccc} d_1 & d_2 & d_3 \\ b_1 & b_2 & b_3\\ c_1 & c_2 & c_3\end {array} \right |$
Condition for two lines to be coplanar is $\left| \begin {array} {ccc} x_1-x_2 & y_1-y_2 & z_1-z_2 \\ l_1 & m_1 & n_1\\ l_2 & m_2 & n_2\end {array} \right |=0$
where $\overrightarrow a_1=(x_1,y_1,z_1),\:\:\overrightarrow a_2=(x_2,y_2,z_2),$
$\overrightarrow b_1=(l_1,m_1,n_1)\:\:and\:\:\overrightarrow b_2=(l_2,m_2,n_2)$
$\Rightarrow\: \left| \begin {array} {ccc} x_1 & y_1 & z_1 \\ l_1 & m_1 & n_1\\ l_2 & m_2 & n_2\end {array} \right | -\left| \begin {array} {ccc} x_2 & y_2 & z_2 \\ l_1 & m_1 & n_1\\ l_2 & m_2 & n_2\end {array} \right |=0$
$\Rightarrow\:[\overrightarrow a_1\:\overrightarrow b_1\:\overrightarrow b_2]-[\overrightarrow a_1\:\overrightarrow b_1\:\overrightarrow b_2]=0$
or $\Rightarrow\:[\overrightarrow a_1\:\overrightarrow b_1\:\overrightarrow b_2]=[\overrightarrow a_1\:\overrightarrow b_1\:\overrightarrow b_2]=0$