Intersection of lines and distancesAQA A-Level Further Maths: Subtopic test

1. 1
   The line $l_1$ has equation $\mathbf r=\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}+\lambda\begin{pmatrix} 1 \\ -1 \\ 2 \end{pmatrix}$ and the line $l_2$ has equation $\mathbf r=\begin{pmatrix} 1 \\ -1 \\ 4 \end{pmatrix}+\mu\begin{pmatrix} 2 \\ 1 \\ 3 \end{pmatrix}$.

   (a)

   Which pair of equations comes from equating the $x$- and $y$-components of $l_1$ and $l_2$?

   [1 mark]

   • A$1+\lambda=2\mu$ and $2-\lambda=\mu$
   • B$1+\lambda=1+2\mu$ and $2+\lambda=-1+\mu$
   • C$1+2\lambda=1+\mu$ and $2+\lambda=-1+2\mu$
   • D$1+\lambda=1+2\mu$ and $2-\lambda=-1+\mu$

   (b)

   Which values of $\lambda$ and $\mu$ satisfy both the $x$ and $y$ equations?

   [1 mark]

   • A$\lambda=2,\ \mu=1$
   • B$\lambda=1,\ \mu=\frac12$
   • C$\lambda=4,\ \mu=2$
   • D$\lambda=0,\ \mu=0$

   (c)

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

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The line $l_3$ has equation $\mathbf r=\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}+\lambda\begin{pmatrix} 1 \\ -1 \\ 2 \end{pmatrix}$ and the line $l_4$ has equation $\mathbf r=\begin{pmatrix} 1 \\ -1 \\ 5 \end{pmatrix}+\mu\begin{pmatrix} 2 \\ 1 \\ 3 \end{pmatrix}$.

   (a)

   Which statement about the directions of $l_3$ and $l_4$ is correct?

   [1 mark]

   • AThe lines are parallel, because both direction vectors have a positive first component.
   • BThe lines are not parallel, because $(1,-1,2)$ is not a multiple of $(2,1,3)$.
   • CThe lines are parallel, because both pass through points with $x=1$.
   • DThe lines are not parallel, because they have different base points.

   (b)

   The $x$ and $y$ equations give $\lambda=2$ and $\mu=1$. What are the $z$-coordinates on $l_3$ and $l_4$ for these values?

   [1 mark]

   • A$7$ on $l_3$ and $7$ on $l_4$
   • B$9$ on $l_3$ and $8$ on $l_4$
   • C$7$ on $l_3$ and $8$ on $l_4$
   • D$5$ on $l_3$ and $8$ on $l_4$

   (c)

   Explain why $l_3$ and $l_4$ are skew lines.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The line $l$ has equation $\mathbf r=\begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}+\lambda\begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix}$. The point $P$ has coordinates $(4,3,2)$.

   (a)

   Find the coordinates of the point $F$ on $l$ for which $PF$ is perpendicular to $l$.

   [3 marks]

   (b)

   Find the perpendicular distance from $P$ to $l$. The point $Q$ is the reflection of $P$ in $l$. Find the coordinates of $Q$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Two straight pipes in a factory are modelled, relative to an origin $O$ at a corner of the floor, by the lines $\mathbf r=\lambda\begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix}$ (pipe $A$) and $\mathbf r=\begin{pmatrix} 1 \\ -1 \\ 2 \end{pmatrix}+\mu\begin{pmatrix} 2 \\ 1 \\ -2 \end{pmatrix}$ (pipe $B$). Distances are in metres.

   (a)

   Show that the two pipes are skew lines.

   [6 marks]

   (b)

   Find the shortest distance between the two pipes.

   [6 marks]

   Total for question 4: 12 marks

End of questions