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

In the plane, determine the value of $k$ such that the lines $L_1: (0,1)+t(2,k)$ and $L_2: (1,0)+s(1,3)$ intersect. Find the intersection point.

Two drone paths are $D_1(t)=(5,t,2t+3)$ and $D_2(s)=(2s+1,4,s-1)$. Find $t$ and $s$ if they collide.

Find the value of $m$ for which the lines $L_1: (2,3,0)+t(1,1,m)$ and $L_2: (0,4,1)+s(3,1,2)$ intersect. 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 $Q(s)=(2s+1,4s-3,s)$. Find the values of $t$ and $s$ at which they meet, and the meeting point.

Determine the value of $a$ for which the lines $L_1: (1,2,3)+t(2,a,1)$ and $L_2: (4,1,5)+s(1,3,2)$ intersect. Also find the intersection point.

For what value of $m$ does the line through $A(1,0,2)$ with direction $(m,2,1)$ intersect the line through $B(0,1,1)$ with direction $(1,m,3)$? Find $m$ and the intersection point.

The drones $D_1$ and $D_2$ follow paths $D_1(t)=(rt,2,t+1)$ and $D_2(s)=(2s,2,s)$. Determine the values of $r$, $t$, and $s$ for which the drones collide.

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 $t$ and $s$.

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

Find $m$, $t$, and $s$ such that the lines $L_1:(1,2,3)+t(4,m,6)$ and $L_2:(3,5,7)+s(2,8,m+1)$ intersect.

Find all values of $k$ such that the skew lines $L_1:(1,0,2)+t(3,k,1)$ and $L_2:(0,2,1)+s(1,4,k-1)$ intersect (i.e., the shortest distance is zero).

Given lines $L_1:(2,-1,3)+t(a,b,c)$ and $L_2:(1,1,1)+s(b,c,a)$, find conditions on $a,b,c$ so that $L_1$ and $L_2$ intersect.