1. **Problem Statement:** We analyze two robotic tracks (lines L1 and L2) and two safety barriers (planes \(\Pi_1\) and \(\Pi_2\)) to determine their spatial relationships and distances. --- ### Tracking Collision (Lines) 2. **Given:** - Line L1 passes through \(P_1(1,0,2)\) with direction vector \(\vec{v}_1=\langle 2,-1,3 \rangle\). - Line L2 in symmetric form: \(\frac{x-3}{1} = \frac{y+2}{2} = \frac{z-1}{1}\). 3. **Convert L2 to parametric form:** Let parameter \(t\): \[ x=3+t,\quad y=-2+2t,\quad z=1+t \] Direction vector \(\vec{v}_2=\langle 1,2,1 \rangle\). 4. **(a) Relationship Analysis:** - Check if \(\vec{v}_1\) and \(\vec{v}_2\) are parallel by testing if one is scalar multiple of the other. - Since \(\vec{v}_1=\langle 2,-1,3 \rangle\) and \(\vec{v}_2=\langle 1,2,1 \rangle\), no scalar \(k\) satisfies \(2=k\cdot1\), \(-1=k\cdot2\), and \(3=k\cdot1\) simultaneously. - So, lines are **not parallel**. 5. Check if lines intersect: - Set parametric equations equal: \[ 1+2s=3+t,\quad 0 - s = -2 + 2t,\quad 2 + 3s = 1 + t \] - From first: \(t=1+2s\). - From second: \(-s = -2 + 2t \Rightarrow -s = -2 + 2(1+2s) = -2 + 2 + 4s = 4s\Rightarrow -s=4s \Rightarrow 5s=0 \Rightarrow s=0\). - Then \(t=1+2(0)=1\). - Check third: \(2 + 3(0) = 1 + 1 \Rightarrow 2 = 2\) true. - So lines **intersect** at \(s=0, t=1\). 6. **Answer (a):** Lines L1 and L2 are **intersecting**. --- 7. **(b) Minimum Proximity:** Since lines intersect, shortest distance is \(0\). --- 8. **(c) Computer Science Context:** Distance thresholding ensures robotic arms maintain safe separation to avoid collisions, enabling automated path planning to dynamically adjust trajectories and maintain operational safety. --- ### Barrier Alignment (Planes) 9. **Given:** - \(\Pi_1: 2x - y + 2z = 10\) with normal vector \(\vec{n}_1=\langle 2,-1,2 \rangle\). - \(\Pi_2: x + 3y - z = 5\) with normal vector \(\vec{n}_2=\langle 1,3,-1 \rangle\). 10. **(a) Angle of Intersection:** - Use formula: \[ \cos \theta = \frac{|\vec{n}_1 \cdot \vec{n}_2|}{|\vec{n}_1||\vec{n}_2|} \] - Compute dot product: \[ \vec{n}_1 \cdot \vec{n}_2 = 2\cdot1 + (-1)\cdot3 + 2\cdot(-1) = 2 - 3 - 2 = -3 \] - Magnitudes: \[ |\vec{n}_1| = \sqrt{2^2 + (-1)^2 + 2^2} = \sqrt{4 + 1 + 4} = \sqrt{9} = 3 \] \[ |\vec{n}_2| = \sqrt{1^2 + 3^2 + (-1)^2} = \sqrt{1 + 9 + 1} = \sqrt{11} \] - So: \[ \cos \theta = \frac{|-3|}{3 \sqrt{11}} = \frac{3}{3\sqrt{11}} = \frac{1}{\sqrt{11}} \] - Angle: \[ \theta = \cos^{-1}\left(\frac{1}{\sqrt{11}}\right) \] 11. **(b) Line of Intersection:** - Direction vector \(\vec{d} = \vec{n}_1 \times \vec{n}_2\): \[ \vec{d} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 2 & -1 & 2 \\ 1 & 3 & -1 \end{vmatrix} = \mathbf{i}((-1)(-1) - 2\cdot3) - \mathbf{j}(2\cdot(-1) - 2\cdot1) + \mathbf{k}(2\cdot3 - (-1)\cdot1) \] \[ = \mathbf{i}(1 - 6) - \mathbf{j}(-2 - 2) + \mathbf{k}(6 + 1) = \langle -5, 4, 7 \rangle \] - To find a point on the line, solve system: \[ 2x - y + 2z = 10 \] \[ x + 3y - z = 5 \] - Let \(z = 0\) for simplicity: \[ 2x - y = 10 \] \[ x + 3y = 5 \] - Multiply second by 2: \[ 2x + 6y = 10 \] - Subtract first: \[ (2x + 6y) - (2x - y) = 10 - 10 \Rightarrow 7y = 0 \Rightarrow y=0 \] - Then from \(2x - y = 10\), \(2x = 10 \Rightarrow x=5\). - So point \(P = (5,0,0)\). - Vector equation: \[ \vec{r} = \langle 5,0,0 \rangle + t \langle -5,4,7 \rangle \] 12. **(c) Distance Between Parallel Planes:** - Plane \(\Pi_3: 2x - y + 2z = 25\). - Normals \(\vec{n}_1 = \vec{n}_3 = \langle 2,-1,2 \rangle\) so planes are parallel. - Distance formula between parallel planes: \[ D = \frac{|d_1 - d_2|}{|\vec{n}|} \] - Here, \(d_1 = 10\), \(d_3 = 25\), and \(|\vec{n}|=3\) (from step 10). - So: \[ D = \frac{|10 - 25|}{3} = \frac{15}{3} = 5 \] --- **Final answers:** - (a) Lines L1 and L2 are intersecting. - (b) Shortest distance between lines is 0. - (c) Distance thresholding is critical for collision avoidance in robotics. - (a) Angle between planes \(\theta = \cos^{-1}\left(\frac{1}{\sqrt{11}}\right)\). - (b) Line of intersection: \(\vec{r} = \langle 5,0,0 \rangle + t \langle -5,4,7 \rangle\). - (c) Planes \(\Pi_1\) and \(\Pi_3\) are parallel; distance between them is 5.