Vectors in two dimensionsAQA A-Level Maths: Subtopic test

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

   (a)

   Find $2\mathbf a+\mathbf b$.

   [1 mark]

   • A$\begin{pmatrix}7\\ -8\end{pmatrix}$
   • B$\begin{pmatrix}2\\ 2\end{pmatrix}$
   • C$\begin{pmatrix}5\\ 0\end{pmatrix}$
   • D$\begin{pmatrix}5\\ -8\end{pmatrix}$

   (b)

   Find $\mathbf a-3\mathbf b$.

   [1 mark]

   • A$\begin{pmatrix}6\\ -14\end{pmatrix}$
   • B$\begin{pmatrix}0\\ 10\end{pmatrix}$
   • C$\begin{pmatrix}4\\ -6\end{pmatrix}$
   • D$\begin{pmatrix}3\\ -12\end{pmatrix}$

   (c)

   Find the value of $p$ for which $\mathbf a+p\mathbf b$ is parallel to the vector $\mathbf i$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   $ABCD$ is a parallelogram with $\overrightarrow{AB}=\mathbf a$ and $\overrightarrow{AD}=\mathbf b$.

   (a)

   Which vector is $\overrightarrow{AC}$?

   [1 mark]

   • A$\mathbf a-\mathbf b$
   • B$\mathbf b-\mathbf a$
   • C$\frac12(\mathbf a+\mathbf b)$
   • D$\mathbf a+\mathbf b$

   (b)

   Which vector is $\overrightarrow{BD}$?

   [1 mark]

   • A$\mathbf a+\mathbf b$
   • B$\mathbf b-\mathbf a$
   • C$\mathbf a-\mathbf b$
   • D$-\mathbf a-\mathbf b$

   (c)

   The point $M$ is the midpoint of $DC$. Find $\overrightarrow{AM}$ in terms of $\mathbf a$ and $\mathbf b$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   $\mathbf u=2\mathbf i+5\mathbf j$, $\mathbf v=\lambda\mathbf i-4\mathbf j$ and $\mathbf w=\mathbf i-\mathbf j$, where $\lambda$ is a constant.

   (a)

   Given that $\mathbf u$ and $\mathbf v$ are parallel, find the value of $\lambda$.

   [3 marks]

   (b)

   Find the values of the constants $s$ and $t$ such that $s\mathbf u+t\mathbf w=4\mathbf i+17\mathbf j$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   In triangle $PQR$, $\overrightarrow{PQ}=\mathbf a$ and $\overrightarrow{PR}=\mathbf b$. The point $M$ is the midpoint of $QR$, and the point $X$ lies on $PR$ so that $PX:XR=2:1$.

   (a)

   Find, in terms of $\mathbf a$ and $\mathbf b$, in their simplest forms: (i) $\overrightarrow{QR}$, (ii) $\overrightarrow{PM}$, (iii) $\overrightarrow{QX}$.

   [6 marks]

   (b)

   The point $N$ is the midpoint of $PQ$. (i) Show that $\overrightarrow{NM}=\frac12\mathbf b$. (ii) Hence show that $NM$ is parallel to $XR$, and find the ratio $NM:XR$.

   [6 marks]

   Total for question 4: 12 marks

End of questions