Further vectorsAQA A-Level Further Maths: Topic test

1. 1
   The line $l$ has Cartesian equation $\dfrac{x-2}{3}=\dfrac{y+1}{1}=\dfrac{z-4}{-2}$.

   (a)

   Which vector is a direction vector of $l$?

   [1 mark]

   • A$\begin{pmatrix}2\\-1\\4\end{pmatrix}$
   • B$\begin{pmatrix}3\\1\\-2\end{pmatrix}$
   • C$\begin{pmatrix}3\\1\\2\end{pmatrix}$
   • D$\begin{pmatrix}-3\\-1\\-2\end{pmatrix}$

   (b)

   Which point lies on $l$ and has $z$ coordinate $0$?

   [1 mark]

   • A$(2,-1,0)$
   • B$(5,0,2)$
   • C$(-4,-3,0)$
   • D$(8,1,0)$

   (c)

   Determine whether the point $B(-1,-2,6)$ lies on $l$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The vectors $\mathbf u=\begin{pmatrix}2\\1\\-2\end{pmatrix}$ and $\mathbf w=\begin{pmatrix}1\\-4\\ k\end{pmatrix}$ are given, where $k$ is a constant.

   (a)

   Which expression gives $\mathbf u\cdot\mathbf w$?

   [1 mark]

   • A$-2-2k$
   • B$6-2k$
   • C$-2-k$
   • D$-2+2k$

   (b)

   Given that $k=2$, what is the angle between $\mathbf u$ and $\mathbf w$?

   [1 mark]

   • A$64.1^\circ$
   • B$95.5^\circ$
   • C$115.9^\circ$
   • D$98.4^\circ$

   (c)

   Find the value of $k$ for which $\mathbf u$ and $\mathbf w$ are perpendicular.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Two cables in a stadium roof are modelled by the lines $l_1$ with equation $\mathbf r=\begin{pmatrix}2\\0\\1\end{pmatrix}+\lambda\begin{pmatrix}1\\2\\-1\end{pmatrix}$ and $l_2$ with equation $\mathbf r=\begin{pmatrix}3\\5\\-3\end{pmatrix}+\mu\begin{pmatrix}1\\-1\\2\end{pmatrix}$.

   (a)

   Show that $l_1$ and $l_2$ intersect and find the coordinates of the point of intersection.

   [3 marks]

   (b)

   Find the acute angle between $l_1$ and $l_2$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The points $P(1,1,0)$, $Q(3,0,1)$ and $R(2,3,2)$ lie in a plane $\Pi$. The line $l$ has equation $\mathbf r=\begin{pmatrix}-1\\2\\0\end{pmatrix}+t\begin{pmatrix}1\\2\\1\end{pmatrix}$.

   (a)

   Use a vector product to find a Cartesian equation of $\Pi$, and find the area of triangle $PQR$.

   [6 marks]

   (b)

   Find the coordinates of the point where $l$ meets $\Pi$, and the acute angle between $l$ and $\Pi$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The line $m$ passes through the points $C(1,4,-2)$ and $D(5,2,0)$.

   (a)

   Which is a vector equation of $m$?

   [1 mark]

   • A$\mathbf r=\begin{pmatrix}1\\4\\-2\end{pmatrix}+\lambda\begin{pmatrix}6\\6\\-2\end{pmatrix}$
   • B$\mathbf r=\begin{pmatrix}5\\2\\0\end{pmatrix}+\lambda\begin{pmatrix}1\\4\\-2\end{pmatrix}$
   • C$\mathbf r=\begin{pmatrix}1\\4\\-2\end{pmatrix}+\lambda\begin{pmatrix}2\\-1\\1\end{pmatrix}$
   • D$\mathbf r=\begin{pmatrix}1\\4\\-2\end{pmatrix}+\lambda\begin{pmatrix}4\\2\\2\end{pmatrix}$

   (b)

   Which of these points lies on $m$?

   [1 mark]

   • A$(7,1,1)$
   • B$(7,-1,1)$
   • C$(3,3,-2)$
   • D$(-3,6,-3)$

   (c)

   Write down the Cartesian equation of $m$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The points $U(1,3,-2)$, $V(4,1,0)$ and $W(2,5,k)$ are given, where $k$ is a constant.

   (a)

   What is $\overrightarrow{VU}$?

   [1 mark]

   • A$\begin{pmatrix}3\\-2\\2\end{pmatrix}$
   • B$\begin{pmatrix}5\\4\\-2\end{pmatrix}$
   • C$\begin{pmatrix}-3\\-2\\-2\end{pmatrix}$
   • D$\begin{pmatrix}-3\\2\\-2\end{pmatrix}$

   (b)

   Angle $UVW$ is $90^\circ$. What is the value of $k$?

   [1 mark]

   • A$-7$
   • B$7$
   • C$14$
   • D$3.5$

   (c)

   Given that $k=0$, find angle $UVW$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A plane $\Pi$ has equation $2x-3y+6z=14$ and a point $A$ has coordinates $(4,1,5)$.

   (a)

   Find the perpendicular distance from $A$ to $\Pi$.

   [3 marks]

   (b)

   Find the coordinates of the foot of the perpendicular from $A$ to $\Pi$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   Plane $\Pi_1$ has equation $2x-2y+z=3$ and plane $\Pi_2$ has equation $3x+4z=10$. The point $(2,1,1)$ lies on both planes.

   (a)

   Find the acute angle between $\Pi_1$ and $\Pi_2$, and the perpendicular distance from the origin to $\Pi_1$.

   [6 marks]

   (b)

   Use a vector product to find a vector equation and the Cartesian equation of the line of intersection of $\Pi_1$ and $\Pi_2$.

   [6 marks]

   Total for question 8: 12 marks

End of questions