3.12 Vector applications to kinematicsIB Maths: Applications and Interpretation HL: Mind map

Exam questions on 3.12 Vector applications to kinematics

1. A drone starts at the point with position vector $\mathbf{r}_0=\begin{pmatrix} 2 \\ 5 \end{pmatrix}$ m and moves with constant velocity $\mathbf{v}=\begin{pmatrix} 3 \\ -4 \end{pmatrix}$ m s$^{-1}$. Its position at time $t$ seconds is $\mathbf{r}=\mathbf{r}_0+t\mathbf{v}$.
   Find the time at which the drone crosses the $x$-axis, and its $x$-coordinate at that time.2 marks
2. Two boats, $A$ and $B$, move with constant velocities. Relative to a harbour at the origin (distances in km), the positions at time $t$ hours after noon are $\mathbf{r}_A=\begin{pmatrix} 1 \\ 2 \end{pmatrix}+t\begin{pmatrix} 4 \\ 1 \end{pmatrix}$ and $\mathbf{r}_B=\begin{pmatrix} 9 \\ -1 \end{pmatrix}+t\begin{pmatrix} -2 \\ 4 \end{pmatrix}$.
   Use your GDC to find the times when the boats are exactly 5 km apart.2 marks
3. Two drones, $D_1$ and $D_2$, fly in a large hall. With the origin at one corner of the floor (distances in metres), their positions $t$ seconds after launch are $\mathbf{r}_1=\begin{pmatrix} 2 \\ -1 \\ 4 \end{pmatrix}+t\begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}$ and $\mathbf{r}_2=\begin{pmatrix} -1 \\ 9 \\ -1 \end{pmatrix}+t\begin{pmatrix} 3 \\ -2 \\ 1 \end{pmatrix}$.
   Show that the paths of the two drones intersect, and find the coordinates of the point of intersection.3 marks