Question Type 5: Finding the value of parameters such that two trajectories or objects would intersect Exercises

Consider the lines $L_1$ and $L_2$ defined by:

$L_1: \mathbf{r} = \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix} + t \begin{pmatrix} 3 \\ k \\ 1 \end{pmatrix}$

$L_2: \mathbf{r} = \begin{pmatrix} 0 \\ 2 \\ 1 \end{pmatrix} + s \begin{pmatrix} 1 \\ 4 \\ k - 1 \end{pmatrix}$

Find all values of $k$ such that the lines intersect.

Given the lines $L_1: (1, -1, 2) + t(2, m, 4)$ and $L_2: (3, 0, 5) + s(1, 2, m)$, determine all values of $m$ for which they intersect, and find the corresponding values of $t$ and $s$.

Find the value of $m$ for which the lines $L_1$ and $L_2$ intersect, where

$L_1: \mathbf{r} = \begin{pmatrix} 2 \\ 3 \\ 0 \end{pmatrix} + t \begin{pmatrix} 1 \\ 1 \\ m \end{pmatrix}$

$L_2: \mathbf{r} = \begin{pmatrix} 0 \\ 4 \\ 1 \end{pmatrix} + s \begin{pmatrix} 3 \\ 1 \\ 2 \end{pmatrix}$

Also find the parameters $t$ and $s$ at the intersection and the intersection point.

Two particles move along paths $P(t)=(t, 2t+1, 3t-2)$ and $Q(s)=(2s+1, 4s-3, s)$. Determine whether the paths intersect and, if so, find the values of $t$ and $s$ at which they meet and the coordinates of the meeting point.

Determine the value of $a$ for which the lines $L_1: \mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} + t \begin{pmatrix} 2 \\ a \\ 1 \end{pmatrix}$ and $L_2: \mathbf{r} = \begin{pmatrix} 4 \\ 1 \\ 5 \end{pmatrix} + s \begin{pmatrix} 1 \\ 3 \\ 2 \end{pmatrix}$ intersect. Also find the coordinates of the intersection point.

Find all possible values of $m$, $t$, and $s$ such that the lines $L_1: (1, 2, 3) + t(4, m, 6)$ and $L_2: (3, 5, 6) + s(2, 8, m+1)$ intersect.

The paths of two drones, $D_1$ and $D_2$, are given by $D_1(t) = (rt, 2, t + 1)$ and $D_2(s) = (2s, 2, s)$ for $t, s ∈ ℝ$.

Determine the expressions for $t$ and $s$ in terms of $r$ for which the paths of the drones intersect, and state the necessary restriction on $r$.

Two drones follow paths defined by the vector equations $D_1(t) = (5, t, 2t+3)$ and $D_2(s) = (2s+1, 4, s-1)$.

Determine whether the drones collide.

Do the trajectories $L_1(t) = (t^2, t, 1)$ and $L_2(s) = (1, 2s, s^2)$ intersect? If so, find the values of $t$ and $s$ and the coordinates of the intersection point.

In the plane, determine the value of $k$ such that the lines $L_1: \begin{pmatrix} 0 \\ 1 \end{pmatrix} + t \begin{pmatrix} 2 \\ k \end{pmatrix}$ and $L_2: \begin{pmatrix} 1 \\ 0 \end{pmatrix} + s \begin{pmatrix} 1 \\ 3 \end{pmatrix}$ intersect. Find the intersection point in terms of $k$.

Given the lines $L_1$ and $L_2$ defined by:

$L_1: \mathbf{r} = \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix} + t \begin{pmatrix} a \\ b \\ c \end{pmatrix}$ $L_2: \mathbf{r} = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} + s \begin{pmatrix} b \\ c \\ a \end{pmatrix}$

find a condition on $a, b, c$ such that the lines $L_1$ and $L_2$ intersect.