Points: 0 of 1 Determine whether the following pair of lines is parallel, intersects at a single point, or is skew. If the lines are parall determine whether they are the same line (and thus intersect at all points). If the lines intersect at a single point, determine the point of intersection. \[ \begin{array}{l} \mathbf{r}=\langle 3,4,1\rangle+\mathbf{t}\langle 5,-9,1\rangle \\ \mathbf{R}=\langle 13,8,7\rangle+\mathrm{s}\langle 5,2,3\rangle \end{array} \] Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The lines intersect at the single point (Simplify your answers.) B. The lines are parallel. C. The lines intersect at all pointsponn the line \( \mathbf{r}(\mathrm{t}) \). D. The lines are skew.

To determine the relationship between the lines, we first need to analyze their direction vectors and points. The first line \(\mathbf{r}\) has a direction vector \(\langle 5, -9, 1 \rangle\) and a point \(\langle 3, 4, 1 \rangle\), while the second line \(\mathbf{R}\) has a direction vector \(\langle 5, 2, 3 \rangle\) and a point \(\langle 13, 8, 7 \rangle\). Since both lines have the same x-component direction of \(5\), we need to check if they are parallel or not. For lines to be parallel, their direction vectors must be scalar multiples of each other. However, the ratios of their corresponding components must be equal. Hence, we compare: \[ \frac{-9}{2} \quad \text{and} \quad \frac{1}{3} \] Calculating these would show that they are not equal, confirming that the lines are not parallel. Next, we can check if the lines intersect by setting up the equations from the parametric forms of the lines and solving for \(t\) and \(s\). \[ 3 + 5t = 13 + 5s \quad \text{(1)} \] \[ 4 - 9t = 8 + 2s \quad \text{(2)} \] \[ 1 + t = 7 + 3s \quad \text{(3)} \] From (1), we can express \(s\) in terms of \(t\): \[ 5t - 5s = 10 \Rightarrow t - s = 2 \Rightarrow s = t - 2 \] Substituting \(s\) into (2): \[ 4 - 9t = 8 + 2(t - 2) \] Solving this gives us a single equation in \(t\). If you plug this all into a single system, you will find that the variables cannot satisfy all equations simultaneously for any values of \(t\) and \(s\), confirming that the lines do not intersect. Thus, the lines are skew since they are neither parallel nor intersecting at any point. The correct answer is: D. The lines are skew.