IB Mathematics Applications & Interpretation Question Type 3: Determining Whether Two Objects Intersect for Their Given Trajectory Exercises

For the moving points $P_1(t)=\left(\begin{array}{c}2\\ 1\\ 4\end{array}\right)+t\left(\begin{array}{c}5\\ 9\\ 1\end{array}\right)$ and $P_2(t)=\left(\begin{array}{c}1\\ 0\\ 1\end{array}\right)+t\left(\begin{array}{c}8\\ 7\\ 2\end{array}\right)$, 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 ${\mathbf{r}}_1=\left(\begin{array}{c}1\\ 2\\ 3\end{array}\right)+t\left(\begin{array}{c}2\\ 5\\ 7\end{array}\right)$ and ${\mathbf{r}}_2=\left(\begin{array}{c}3\\ 4\\ 5\end{array}\right)+s\left(\begin{array}{c}4\\ 3\\ 1\end{array}\right)$ intersect.

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

Find the value of $m$ for which the lines ${\mathbf{r}}_1=\left(\begin{array}{c}5\\ 2\\ 1\end{array}\right)+t\left(\begin{array}{c}m\\ 3\\ 4\end{array}\right)$ and ${\mathbf{r}}_2=\left(\begin{array}{c}2\\ 5\\ 3\end{array}\right)+s\left(\begin{array}{c}4\\ -1\\ 2\end{array}\right)$ 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 ${\mathbf{r}}_1=\left(\begin{array}{c}2\\ 3\\ 0\end{array}\right)+t\left(\begin{array}{c}1\\ 1\\ m\end{array}\right)$ and ${\mathbf{r}}_2=\left(\begin{array}{c}0\\ 4\\ 1\end{array}\right)+s\left(\begin{array}{c}3\\ 1\\ 2\end{array}\right)$ intersect.

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

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

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

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