Vectors and translation vectorsIB MYP Maths Extended: Subtopic test

1. 1
   The vectors $\mathbf p$ and $\mathbf q$ are given by $\mathbf p=\begin{pmatrix}3\\ -2\end{pmatrix}$ and $\mathbf q=\begin{pmatrix}-1\\ 4\end{pmatrix}$.

   (a)

   Find $\mathbf p+2\mathbf q$.

   [1 mark]

   • A$\begin{pmatrix}1\\ 6\end{pmatrix}$
   • B$\begin{pmatrix}2\\ 2\end{pmatrix}$
   • C$\begin{pmatrix}5\\ -10\end{pmatrix}$
   • D$\begin{pmatrix}1\\ 2\end{pmatrix}$

   (b)

   Find the magnitude $|\mathbf p-\mathbf q|$.

   [1 mark]

   • A$10$
   • B$\sqrt8$
   • C$\sqrt{13}+\sqrt{17}$
   • D$\sqrt{52}$

   (c)

   The point $A(2,5)$ is translated by $\mathbf p$ and then by $\mathbf q$. Find the coordinates of its final image.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   $ABCD$ is a parallelogram, with the vertices in that order, and $A(1,1)$, $B(6,2)$ and $C(9,6)$.

   (a)

   Find the vector $\overrightarrow{BC}$.

   [1 mark]

   • A$\begin{pmatrix}-3\\ -4\end{pmatrix}$
   • B$\begin{pmatrix}3\\ 4\end{pmatrix}$
   • C$\begin{pmatrix}15\\ 8\end{pmatrix}$
   • D$\begin{pmatrix}9\\ 6\end{pmatrix}$

   (b)

   Find the coordinates of $D$.

   [1 mark]

   • A$(14,7)$
   • B$(-2,-3)$
   • C$(4,5)$
   • D$(5,4)$

   (c)

   Find the exact length of the diagonal $[AC]$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A drone starts at the point $S(2,-1)$ on a map grid, where one unit on the grid is 1 km. It makes a translation $\begin{pmatrix}4\\ 3\end{pmatrix}$ and then a translation $\begin{pmatrix}-7\\ 2\end{pmatrix}$ to reach the point $T$.

   (a)

   Find the coordinates of $T$, and write down the single translation that takes $S$ directly to $T$.

   [3 marks]

   (b)

   The drone now flies in a straight line from $T$ back to $S$ at a constant speed of 30 km/h. Find the translation from $T$ to $S$, the distance flown, and the time taken, in minutes, to 3 significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A coastguard station $O$ is at the origin of a map grid, where one unit is 1 km. Boat $A$ is at the point with $\overrightarrow{OA}=\begin{pmatrix}4\\ 2\end{pmatrix}$ and boat $B$ is at the point with $\overrightarrow{OB}=\begin{pmatrix}-2\\ 6\end{pmatrix}$. The point $M$ is the midpoint of $[AB]$.

   (a)

   (i) Find the vector $\overrightarrow{AB}$.
   (ii) Hence find the vector $\overrightarrow{OM}$.
   (iii) Find the exact distance from $O$ to $M$.

   [6 marks]

   (b)

   A lifeboat leaves $O$ and sails at a constant 15 km/h in a straight line to $M$, and then in a straight line from $M$ to $A$. Coastguard rules say that it must reach $A$ within 30 minutes of leaving $O$.
   Justify whether the rule is met. Give times in minutes to 3 significant figures.

   [6 marks]

   Total for question 4: 12 marks

End of questions