# Step-by-Step Solution Let's break the problem into two main parts: 1. **Equation of the straight-line path from \(A\) to \(B\)** 2. **Shortest distance from point \(C\) to this path** Given coordinates: - \(A(, 3, -4)\) - \(B(-3, 1, 2)\) - \(C(-2, 3, 2)\) --- ## 1. Equation of the Straight-Line Path \(AB\) A line passing through point \(A(x_1, y_1, z_1)\) and parallel to vector \( \vec{AB} \) can be written as: \[ \vec{r}(t) = \vec{A} + t(\vec{B} - \vec{A}) \] Calculate \( \vec{AB} \): \[ \vec{AB} = B - A = \begin{pmatrix} -3 - \\ 1 - 3 \\ 2 - (-4) \end{pmatrix} = \begin{pmatrix} -3 \\ -2 \\ 6 \end{pmatrix} \] So, the vector equation is: \[ \vec{r}(t) = (, 3, -4) + t(-3, -2, 6) \] or in parametric form: \[ \begin{cases} x = -3t \\ y = 3 - 2t \\ z = -4 + 6t \end{cases} , \quad t \in \mathbb{R} \] --- ## 2. Shortest Distance from \(C\) to Line \(AB\) The shortest distance \(d\) from a point \(C\) to a line through \(A\) with direction vector \(\vec{AB}\) is: \[ d = \frac{|\vec{AB} \times \vec{AC}|}{|\vec{AB}|} \] Where: - \( \vec{AC} = C - A = (-2 - , 3 - 3, 2 - (-4)) = (-2, , 6) \) ### Step 1: Compute \( \vec{AB} \times \vec{AC} \) \[ \vec{AB} = (-3, -2, 6),\quad \vec{AC} = (-2, , 6) \] The cross product is: \[ \vec{AB} \times \vec{AC} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ -3 & -2 & 6 \\ -2 & & 6 \\ \end{vmatrix} \] \[ = \mathbf{i}((-2) \cdot 6 - 6 \cdot ) - \mathbf{j}((-3) \cdot 6 - 6 \cdot (-2)) + \mathbf{k}((-3) \cdot - (-2) \cdot (-2)) \] \[ = \mathbf{i}(-12 - ) - \mathbf{j}(-18 + 12) + \mathbf{k}( - 4) \] \[ = \mathbf{i}(-12) - \mathbf{j}(-6) + \mathbf{k}(-4) \] \[ = (-12, 6, -4) \] ### Step 2: Compute Magnitudes \[ |\vec{AB} \times \vec{AC}| = \sqrt{(-12)^2 + 6^2 + (-4)^2} = \sqrt{144 + 36 + 16} = \sqrt{196} = 14 \] \[ |\vec{AB}| = \sqrt{(-3)^2 + (-2)^2 + 6^2} = \sqrt{9 + 4 + 36} = \sqrt{49} = 7 \] ### Step 3: Compute Distance \[ d = \frac{14}{7} = 2 \] --- ## **Final Answers** ### **Equation of the Path from \(A\) to \(B\):** \[ \boxed{ \begin{cases} x = -3t \\ y = 3 - 2t \\ z = -4 + 6t \end{cases} ,\quad t \in \mathbb{R} } \] ### **Shortest Distance from \(C\) to Path \(AB\):** \[ \boxed{2} \] This means the closest the robot gets to the safety sensor at \(C\) is **2 units**.