In equation (1), \( - 4\) for \({x_1}, - 6\) for \({y_1},1\) for \({z_1}, - 2\) for \({x_2},0\) for \({y_2}\), and \( - 3\) for \({z_2}\).
\(\begin{array}{l}{{\rm{v}}_1} = \langle ( - 2 - ( - 4)),(0 - ( - 6)),( - 3 - 1)\rangle \\{{\rm{v}}_1} = \langle 2,6, - 4\rangle \end{array}\)
In equation (1), 10 for \({x_1},18\) for \({y_1},4\) for \({z_1},5\) for \({x_2},3\) for \({y_2}\), and 14 for \({z_2}\).
\(\begin{array}{l}{v_2} = \langle (5 - 10),(3 - 18),(14 - 4)\rangle \\{v_2} = \langle - 5, - 15,10\rangle \end{array}\)
The direction vector \({v_2}\) is also written as.
\(\begin{array}{l}{{\rm{v}}_2} = \left( { - \frac{5}{2}} \right)\langle 2,6, - 4\rangle \\{{\rm{v}}_2} = \left( { - \frac{5}{2}} \right){{\rm{v}}_1}\end{array}\)
As the direction vector of line through the points \((10,18,4)\) and \((5,3,14)\) is multiple of the direction vector of line through the points \(( - 4, - 6,1)\) and \(( - 2,0, - 3)\), the two lines are in parallel.
Thus, the line through the points \(( - 4, - 6,1)\) and \(( - 2,0, - 3)\) is parallel to the line through the points \((10,18,4)\) and \((5,3,14)\).
