3.10 Vectors: concepts and algebraIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   Vectors $\mathbf{p}=\begin{pmatrix}2 \\ -1 \\ 2\end{pmatrix}$ and $\mathbf{q}=\begin{pmatrix}1 \\ 3 \\ -4\end{pmatrix}$.

   (a)

   Find $2\mathbf{p}-\mathbf{q}$.

   [1 mark]

   • A$\begin{pmatrix}5 \\ 1 \\ 0\end{pmatrix}$
   • B$\begin{pmatrix}3 \\ -5 \\ 0\end{pmatrix}$
   • C$\begin{pmatrix}-3 \\ 5 \\ -8\end{pmatrix}$
   • D$\begin{pmatrix}3 \\ -5 \\ 8\end{pmatrix}$

   (b)

   Find $|\mathbf{p}|$.

   [1 mark]

   • A$9$
   • B$3$
   • C$\sqrt{3}$
   • D$\sqrt{26}$

   (c)

   Find the vector of magnitude $12$ that has the same direction as $\mathbf{p}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Relative to an origin $O$, the points $A$ and $B$ have position vectors $\overrightarrow{OA}=\begin{pmatrix}1 \\ 4 \\ -2\end{pmatrix}$ and $\overrightarrow{OB}=\begin{pmatrix}3 \\ 7 \\ 4\end{pmatrix}$.

   (a)

   Find $\overrightarrow{AB}$.

   [1 mark]

   • A$\begin{pmatrix}-2 \\ -3 \\ -6\end{pmatrix}$
   • B$\begin{pmatrix}4 \\ 11 \\ 2\end{pmatrix}$
   • C$\begin{pmatrix}2 \\ 3 \\ 6\end{pmatrix}$
   • D$\begin{pmatrix}2 \\ 3 \\ -6\end{pmatrix}$

   (b)

   Find the distance $AB$.

   [1 mark]

   • A$7$
   • B$11$
   • C$49$
   • D$\sqrt{11}$

   (c)

   Find the position vector of the midpoint $M$ of $[AB]$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A drone starts at the origin $O$, which is at ground level. The unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ point east, north and vertically upwards, and distances are in metres. The drone flies at constant speed $12$ m s$^{-1}$ in the direction of the vector $\mathbf{i}+2\mathbf{j}+2\mathbf{k}$.

   (a)

   Find the velocity vector of the drone.

   [3 marks]

   (b)

   (i) Find the position vector of the drone after $5$ seconds.
   (ii) Find the distance of the drone from $O$ after $5$ seconds.
   (iii) Find the time at which the drone is $60$ m above the ground.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Three forces act on a ship. In kN, with $\mathbf{i}$ pointing east and $\mathbf{j}$ pointing north, $\mathbf{F}_1=6\mathbf{i}+8\mathbf{j}$, $\mathbf{F}_2=-2\mathbf{i}+5\mathbf{j}$ and $\mathbf{F}_3=k\mathbf{i}-3\mathbf{j}$, where $k$ is a constant.

   (a)

   (i) Find the resultant of $\mathbf{F}_1$ and $\mathbf{F}_2$.
   (ii) Find the magnitude of this resultant.
   (iii) Find the value of $k$ for which the resultant of all three forces acts due north.
   (iv) Find the magnitude of the resultant of all three forces for this value of $k$.

   [6 marks]

   (b)

   (i) A fourth force $\mathbf{F}_4$ has magnitude $25$ kN and acts in the same direction as $\mathbf{F}_1$. Find $\mathbf{F}_4$.
   (ii) A fifth force $\mathbf{F}_5=m\mathbf{i}+12\mathbf{j}$ is parallel to $\mathbf{F}_1$. Find the value of $m$.
   (iii) Find the unit vector in the direction of $\mathbf{F}_1+\mathbf{F}_2$.

   [6 marks]

   Total for question 4: 12 marks

End of questions