Determining Collision of 3D Moving Particles
Two particles move in space. Their positions are given by the vector equations: \(\mathbf{r}_1(t) = (t, 2t, 3t)\) \(\mathbf{r}_2(t) = (1, -1, 4) + t\,(0, 3, -1)\) Determine if these particles collide. If a collision occurs, find the time \(t\) at which the collision happens and the coordinates of the collision point.
3D Trajectory Analysis Using Vector Parametrics
Consider a particle moving in 3D space with position vector \(\mathbf{r}(t) = (3t + 2, t^2 – 1, 4 – t)\) where \(t\) is the time in seconds. (a) Express the motion in terms of its directional components and determine the velocity vector \(\mathbf{v}(t)\). (b) Determine if there exists a time \(t\) when the particle […]
Part b
Determine if there exists a time \(t\) when \(\mathbf{r}(t)\) equals the position \((11, 8, 2)\). If such a time exists, find \(t\), providing justification for each coordinate equality.
Modeling Constrained Vector Path
Consider a scenario where an autonomous drone must navigate a planar environment while avoiding a designated obstacle region. The drone’s path is modeled as a parametric vector function. Answer the following parts regarding the path’s parameterization and the condition required for obstacle avoidance.
Part b
Given an obstacle represented by the region between the parallel lines \(y = \tan(\alpha)x + d\) and \(y = \tan(\alpha)x + d + L\) (with \(L > 0\)), determine the condition on \(\theta\) such that the drone’s path does not intersect the obstacle. Assume the drone starts from a position below the obstacle.
Intersection of Flight Paths via Vector Equations
A navigation officer is monitoring two aircraft and needs to determine if their flight paths intersect to assess potential collision risks. The flight path of the first aircraft is described by the vector equation: \( \vec{r}_1(t) = \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix} + t \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix} \), where […]