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

For the moving points $P_1(t) = \begin{pmatrix} 2 \\ 1 \\ 4 \end{pmatrix} + t\begin{pmatrix} 5 \\ 9 \\ 1 \end{pmatrix}$ and $P_2(t) = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} + t\begin{pmatrix} 8 \\ 7 \\ 2 \end{pmatrix}$, find the time $t$ at which the points are closest to each other.

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.

Determine whether the lines $\mathbf{r}_1 = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} + t \begin{pmatrix} 2 \\ 5 \\ 7 \end{pmatrix}$ and $\mathbf{r}_2 = \begin{pmatrix} 3 \\ 4 \\ 5 \end{pmatrix} + s \begin{pmatrix} 4 \\ 3 \\ 1 \end{pmatrix}$ intersect.

Find the time $t$ at which $P_1(t) = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} + t \begin{pmatrix} 2 \\ 0 \\ 1 \end{pmatrix}$ and $P_2(t) = \begin{pmatrix} 0 \\ 2 \\ 3 \end{pmatrix} + t \begin{pmatrix} 1 \\ 3 \\ -1 \end{pmatrix}$ are closest.

Find the value of $m$ for which the lines $\mathbf{r}_1 = \begin{pmatrix} 5 \\ 2 \\ 1 \end{pmatrix} + t \begin{pmatrix} m \\ 3 \\ 4 \end{pmatrix}$ and $\mathbf{r}_2 = \begin{pmatrix} 2 \\ 5 \\ 3 \end{pmatrix} + s \begin{pmatrix} 4 \\ -1 \\ 2 \end{pmatrix}$ intersect.

Find the minimum distance between the points $P_1(t)=(0,0,0)+t(1,2,3)$ and $P_2(t)=(3,1,0)+t(4,-1,2)$.

Find the value of $m$ such that the lines $\mathbf{r}_1 = \begin{pmatrix} 2 \\ 3 \\ 0 \end{pmatrix} + t \begin{pmatrix} 1 \\ 1 \\ m \end{pmatrix}$ and $\mathbf{r}_2 = \begin{pmatrix} 0 \\ 4 \\ 1 \end{pmatrix} + s \begin{pmatrix} 3 \\ 1 \\ 2 \end{pmatrix}$ intersect.

Determine whether the lines with equations $\mathbf{r}_1 = \begin{pmatrix} 2 \\ 1 \\ 4 \end{pmatrix} + t \begin{pmatrix} 5 \\ 9 \\ 1 \end{pmatrix}$ and $\mathbf{r}_2 = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} + s \begin{pmatrix} 8 \\ 7 \\ 2 \end{pmatrix}$ intersect.

Determine the minimum distance between $P_1$ and $P_2$.

Determine whether the lines $\mathbf{r}_1 = \begin{pmatrix} 0 \\ -1 \\ 2 \end{pmatrix} + t \begin{pmatrix} 3 \\ 4 \\ 1 \end{pmatrix}$ and $\mathbf{r}_2 = \begin{pmatrix} 2 \\ 1 \\ 3 \end{pmatrix} + s \begin{pmatrix} 6 \\ 5 \\ 2 \end{pmatrix}$ intersect.

Determine the value of $m$ such that the lines $\mathbf{r}_1 = \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix} + t \begin{pmatrix} 3 \\ m \\ 5 \end{pmatrix}$ and $\mathbf{r}_2 = \begin{pmatrix} 2 \\ 4 \\ 1 \end{pmatrix} + s \begin{pmatrix} 6 \\ 1 \\ 3 \end{pmatrix}$ intersect.