Vectors in two dimensionsEdexcel A-Level Maths: Subtopic test

1. 1
   The vectors $\mathbf{a}$ and $\mathbf{b}$ are given by $\mathbf{a}=5\mathbf{i}-12\mathbf{j}$ and $\mathbf{b}=-\mathbf{i}+3\mathbf{j}$.

   (a)

   Find $|\mathbf{a}|$.

   [1 mark]

   • A$7$
   • B$17$
   • C$13$
   • D$169$

   (b)

   Which expression is the unit vector $\hat{\mathbf{a}}$ in the direction of $\mathbf{a}$?

   [1 mark]

   • A$\frac{1}{13}(5\mathbf{i}-12\mathbf{j})$
   • B$\frac{1}{169}(5\mathbf{i}-12\mathbf{j})$
   • C$\frac{1}{7}(5\mathbf{i}-12\mathbf{j})$
   • D$\frac{1}{13}(-5\mathbf{i}+12\mathbf{j})$

   (c)

   Find the exact value of $|\mathbf{a}+2\mathbf{b}|$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The vector $\mathbf{p}$ is given by $\mathbf{p}=-4\mathbf{i}+4\sqrt{3}\,\mathbf{j}$.

   (a)

   The vector $\mathbf{p}$ makes an angle $\theta$ with the positive $x$-axis, measured anticlockwise, where $0^\circ\le\theta<360^\circ$. Find $\theta$.

   [1 mark]

   • A$60^\circ$
   • B$240^\circ$
   • C$150^\circ$
   • D$120^\circ$

   (b)

   Find $|\mathbf{p}|$.

   [1 mark]

   • A$4+4\sqrt{3}$
   • B$8$
   • C$64$
   • D$4\sqrt{2}$

   (c)

   Find the vector of magnitude $24$ in the same direction as $\mathbf{p}$, giving your answer in terms of $\mathbf{i}$ and $\mathbf{j}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The vectors $\mathbf{a}$ and $\mathbf{b}$ are given by $\mathbf{a}=2\mathbf{i}+3\mathbf{j}$ and $\mathbf{b}=(k+1)\mathbf{i}+(k-2)\mathbf{j}$, where $k$ is a constant.

   (a)

   Given that $\mathbf{b}$ is parallel to $\mathbf{a}$, find the value of $k$.

   [3 marks]

   (b)

   Given now that $k=2$, find the values of the constants $\lambda$ and $\mu$ such that $\lambda\mathbf{a}+\mu\mathbf{b}=5\mathbf{i}-6\mathbf{j}$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A walker travels from $A$ to $B$ and then from $B$ to $C$. Take $\mathbf{i}$ as a unit vector due east and $\mathbf{j}$ as a unit vector due north, with distances in km. $\overrightarrow{AB}=6\mathbf{i}+8\mathbf{j}$ and $\overrightarrow{BC}=-2\mathbf{i}+5\mathbf{j}$.

   (a)

   (i) Find $\overrightarrow{AC}$ and the exact distance $AC$.
   (ii) Find the angle that $\overrightarrow{AC}$ makes with the vector $\mathbf{i}$, to 1 decimal place.
   (iii) Find the unit vector in the direction of $\overrightarrow{AC}$.

   [6 marks]

   (b)

   (i) Find $|\overrightarrow{AB}|$ and write down the unit vector in the direction of $\overrightarrow{AB}$.
   The point $E$ lies on the line through $C$ parallel to $AB$, and $E$ is due north of $A$.
   (ii) Find $\overrightarrow{AE}$.

   [6 marks]

   Total for question 4: 12 marks

End of questions