Question

Determine whether the lines \({L_1}\) and \({L_2}\) are parallel, skew or intersected lines.

Short answer

The two lines \({L_1}\) and \({L_2}\) are parallel lines.

Step-by-step solution

Direction vector of both the lines.

The two lines must be parallel, skew or intersect lines.

If the two lines are parallel, the direction vectors of both the lines are scalar multiples of each other.

The two lines\({L_1}\)and \({L_2}\)are in the form of parametric equations.

The direction vector of line\({L_1}\left( {{{\rm{v}}_1}} \right){\rm{ is }}\langle - 12,9, - 3\rangle .\)

The direction vector of line\({L_2}\left( {{{\rm{v}}_2}} \right){\rm{ is }}\langle 8, - 6,2\rangle .\)

The direction vector of line\({L_{\rm{I}}}\left( {{{\rm{v}}_1}} \right)\)is also written as follows.

\(\begin{array}{l}{{\rm{v}}_1} = - \frac{3}{2}\langle 8, - 6,2\rangle \\ = - \frac{3}{2}{{\rm{v}}_2}\end{array}\)

Two lines are scalar multiple of each other.

From the calculation of direction vectors, it is clear that the two lines are scalar multiples of each other.

Thus, the two lines \({L_1}\) and \({L_2}\) are parallel lines.