IB Mathematics Applications & Interpretation 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:

$$L_1:\mathbf{r}=\left(\begin{array}{c}1\\ 0\\ 2\end{array}\right)+t\left(\begin{array}{c}3\\ k\\ 1\end{array}\right)$$

$$L_2:\mathbf{r}=\left(\begin{array}{c}0\\ 2\\ 1\end{array}\right)+s\left(\begin{array}{c}1\\ 4\\ k-1\end{array}\right)$$

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

$$L_1:\mathbf{r}=\left(\begin{array}{c}2\\ 3\\ 0\end{array}\right)+t\left(\begin{array}{c}1\\ 1\\ m\end{array}\right)$$

$$L_2:\mathbf{r}=\left(\begin{array}{c}0\\ 4\\ 1\end{array}\right)+s\left(\begin{array}{c}3\\ 1\\ 2\end{array}\right)$$

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 $L_1:\mathbf{r}=\left(\begin{array}{c}1\\ 2\\ 3\end{array}\right)+t\left(\begin{array}{c}2\\ a\\ 1\end{array}\right)$ and $L_2:\mathbf{r}=\left(\begin{array}{c}4\\ 1\\ 5\end{array}\right)+s\left(\begin{array}{c}1\\ 3\\ 2\end{array}\right)$ 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 $t,s\in R$.

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 $L_1:\left(\begin{array}{c}0\\ 1\end{array}\right)+t\left(\begin{array}{c}2\\ k\end{array}\right)$ and $L_2:\left(\begin{array}{c}1\\ 0\end{array}\right)+s\left(\begin{array}{c}1\\ 3\end{array}\right)$ intersect. Find the intersection point in terms of $k$.

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

$$L_1:\mathbf{r}=\left(\begin{array}{c}2\\ -1\\ 3\end{array}\right)+t\left(\begin{array}{c}a\\ b\\ c\end{array}\right)$$ $$L_2:\mathbf{r}=\left(\begin{array}{c}1\\ 1\\ 1\end{array}\right)+s\left(\begin{array}{c}b\\ c\\ a\end{array}\right)$$

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