Question Type 3: Determining whether two objects intersect for their given trajectory Exercises

Determine whether the lines with equations $r_1=(2,1,4)+t(5,9,1)$ and $r_2=(1,0,1)+s(8,7,2)$ intersect.

Determine whether the lines $r_1=(1,2,3)+t(2,5,7)$ and $r_2=(3,4,5)+s(4,3,1)$ intersect.

Determine whether the lines $r_1=(0,-1,2)+t(3,4,1)$ and $r_2=(2,1,3)+s(6,5,2)$ intersect.

Using $P_1$ and $P_2$ from question 4, compute the minimum distance between them.

Find the time $t$ at which $P_1(t)=(1,1,1)+t(2,0,1)$ and $P_2(t)=(0,2,3)+t(1,3,-1)$ are closest.

For the moving points $P_1(t)=(2,1,4)+t(5,9,1)$ and $P_2(t)=(1,0,1)+t(8,7,2)$, find the time $t$ at which they are closest.

Using the result of question 5, find the minimum distance between $P_1(t)=(0,0,0)+t(1,2,3)$ and $P_2(t)=(3,1,0)+t(4,-1,2)$.

For the points $P_1(t)=(0,0,0)+t(1,2,3)$ and $P_2(t)=(3,1,0)+t(4,-1,2)$, determine the time $t$ at which the distance between them is minimal.

Find the value of $m$ such that the lines $r_1=(2,3,0)+t(1,1,m)$ and $r_2=(0,4,1)+s(3,1,2)$ intersect.

Find $m$ for which the lines $r_1=(5,2,1)+t(m,3,4)$ and $r_2=(2,5,3)+s(4,-1,2)$ intersect.

Determine $m$ so that $r_1=(1,0,2)+t(3,m,5)$ and $r_2=(2,4,1)+s(6,1,3)$ intersect.