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 skew lines.

Step-by-step solution

Observe the direction vectors of two 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( {{v_1}} \right){\rm{ is }}\langle 2, - 1,3\rangle .\)

The direction vector of line\){L_2}\left( {{v_2}} \right){\rm{ is }}\langle 4, - 2,5\rangle .\)

By observing the direction vectors of two lines \){L_1}\)and\){L_2}\), it is clear that the two lines are not scalar multiples of each other.

\){{\rm{v}}_1} \ne k{{\rm{v}}_2}\)

Therefore, the two line \){L_1}\)and \){L_2}\)are not parallel lines.

Express equations for the line to intersect.

If the two lines have an intersection point, the parametric equations of the two lines must be equal.

The equations are expressed for the lines to be intersected as follows.

\)3 + 2t = 1 + 4s\) …… (1)

\)4 - t = 3 - 2s\) …… (2)

\)1 + 3t = 4 + 5s\) …… (3)

Solve the equation (2) and (3) as follows.

Rearrange the equation (2).

\)\begin{array}{l}4 - t = 3 - 2s\\t = 4 - 3 - 2s\\t = 1 - 2s\end{array}\)

Substitute\)(1 - 2s)\) for\)t\)in equation (3).

\)\begin{array}{l}1 + 3(1 - 2s) = 4 + 5s\\1 + 3 - 6s = 4 + 5s\\11s = 0\\s = 0\end{array}\)

In the expression\)t = 1 - 2s\), substitute\)0\) for\)s.\)

\)\begin{array}{l}t = 1 - 2(0)\\ = 1\end{array}\)

Substitute\)0\) for\)s\) and\)1\) for\)t\) in equation (1).

\)\begin{array}{l}3 + 2(1) = 1 + 4(0)\\5 \ne 1\end{array}\)

From the calculations, it is clear that the two line are not satisfied the condition of intersection of two lines.

Therefore, the two lines are not intersected lines.

As the two lines are neither parallel nor intersected lines, the two lines must be skew lines.

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