Question

Show that the lines $\frac{x+3}{-3}=\frac{y-1}{1}=\frac{z-5}{5}$ and $\frac{x+1}{-1}=\frac{y-2}{2}=\frac{z-5}{5}$ are coplanar.

Answer 1

$\frac{x+3}{-3}=\frac{y-1}{1}=\frac{z-5}{5}$ and $\frac{x+1}{-1}=\frac{y-2}{2}=\frac{z-5}{5}$

If two lines are coplanar then

$\left|\begin{array}{ccc}{x}_1-x_2& y_1-y_2& z_1-z_2\\ l_1& m_1& n_1\\ l_2& m_2& n_2\end{array}\right|=0$

Lets calculate it

$=\left|\begin{array}{ccc}-3+1& 1-2& 5-5\\ -3& 1& 5\\ -1& 2& 5\end{array}\right|=\left|\begin{array}{ccc}-2& -1& 0\\ -3& 1& 5\\ -1& 2& 5\end{array}\right|$

$=-2(-5)+1(-10)$

$=0$

Lines are coplanar.