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 two given lines in 3D space, let's analyze their parametric equations: \[ \begin{aligned} \mathbf{r}(t) &= \langle 3, 4, 1 \rangle + t \langle 5, -9, 1 \rangle, \\ \mathbf{R}(s) &= \langle 13, 8, 7 \rangle + s \langle 5, 2, 3 \rangle. \end{aligned} \] **Step 1: Check for Parallelism** First, we determine if the direction vectors are parallel. The direction vectors are: \[ \mathbf{v} = \langle 5, -9, 1 \rangle, \quad \mathbf{w} = \langle 5, 2, 3 \rangle. \] To be parallel, there must be a scalar \( k \) such that \( \mathbf{v} = k\mathbf{w} \). Comparing the components: \[ \begin{aligned} 5 &= 5k \quad \Rightarrow \quad k = 1, \\ -9 &= 2k \quad \Rightarrow \quad k = -4.5, \\ 1 &= 3k \quad \Rightarrow \quad k = \frac{1}{3}. \end{aligned} \] Since there is no consistent scalar \( k \) that satisfies all components, the lines are **not parallel**. **Step 2: Check for Intersection** To find if the lines intersect, set the parametric equations equal to each other and solve for \( t \) and \( s \): \[ \begin{aligned} 3 + 5t &= 13 + 5s, \quad \text{(1)} \\ 4 - 9t &= 8 + 2s, \quad \text{(2)} \\ 1 + t &= 7 + 3s. \quad \text{(3)} \end{aligned} \] **From Equation (1):** \[ 5t - 5s = 10 \quad \Rightarrow \quad t - s = 2 \quad \Rightarrow \quad t = s + 2. \] **Substitute \( t = s + 2 \) into Equation (3):** \[ (s + 2) - 3s = 6 \quad \Rightarrow \quad -2s + 2 = 6 \quad \Rightarrow \quad -2s = 4 \quad \Rightarrow \quad s = -2. \] **Find \( t \):** \[ t = s + 2 = -2 + 2 = 0. \] **Verify in Equation (2):** \[ 4 - 9(0) = 8 + 2(-2) \quad \Rightarrow \quad 4 = 4. \] The equations are consistent, confirming the lines intersect. **Point of Intersection:** \[ \mathbf{r}(0) = \langle 3, 4, 1 \rangle. \] **Conclusion:** **Answer A** is correct. The lines intersect at the single point \( (3, 4, 1) \). **Answer:** **A. The lines intersect at the single point** \( (3,\ 4,\ 1) \).