Intersections and distances in three dimensionsEdexcel A-Level Further Maths: Subtopic test

1. 1
   The line $l$ has equation $\mathbf{r}=\begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}+t\begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}$ and the plane $\Pi$ has equation $2x-y+3z=14$.

   (a)

   Find the value of $t$ at the point where $l$ meets $\Pi$.

   [1 mark]

   • A$t=2$
   • B$t=-2$
   • C$t=-6$
   • D$t=-\frac{14}{3}$

   (b)

   Find the position vector of the point where $l$ meets $\Pi$.

   [1 mark]

   • A$(3,4,0)$
   • B$(-5,-12,8)$
   • C$(-1,-4,4)$
   • D$(-1,-4,0)$

   (c)

   The point $A$ with position vector $(1,0,2)$ lies on $l$. Find the exact distance from $A$ to the point where $l$ meets $\Pi$.

   [2 marks]

   Total for question 1: 4 marks

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

   (a)

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

   [1 mark]

   • A$4$
   • B$5$
   • C$\frac59$
   • D$\frac53$

   (b)

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

   [1 mark]

   • A$\left(\frac{17}{9},\frac19,\frac{31}{9}\right)$
   • B$\left(\frac{37}{9},-\frac{19}{9},\frac{41}{9}\right)$
   • C$\left(-\frac13,\frac73,\frac73\right)$
   • D$(-7,9,-1)$

   (c)

   Find the position vector of the reflection of $A$ in $\Pi$.

   [2 marks]

   Total for question 2: 4 marks

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

   (a)

   Find the coordinates of the foot of the perpendicular from $P$ to $l$.

   [3 marks]

   (b)

   Find the shortest distance from $P$ to $l$. Hence find the exact area of the triangle $PAB$, where $A$ and $B$ are the points on $l$ with $\lambda=0$ and $\lambda=1$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The lines $l_1$ and $l_2$ have equations $l_1:\ \mathbf{r}=\begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix}+s\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}$ and $l_2:\ \mathbf{r}=\begin{pmatrix} 0 \\ 0 \\ 3 \end{pmatrix}+t\begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}$.

   (a)

   (i) Show that $l_1$ and $l_2$ are skew.
   (ii) Find a vector that is perpendicular to both $l_1$ and $l_2$.

   [6 marks]

   (b)

   (i) Find the shortest distance between $l_1$ and $l_2$.
   (ii) The plane $\Pi$ contains $l_1$ and is parallel to $l_2$. Find a Cartesian equation of $\Pi$, and use the distance from a point to a plane to confirm your answer to (i).

   [6 marks]

   Total for question 4: 12 marks

End of questions