Lines and planes in three dimensionsEdexcel International A Level Further Maths: Subtopic test

1. 1
   The plane $\Pi$ has equation $\mathbf{r}\cdot(2\mathbf{i}-\mathbf{j}+2\mathbf{k})=6$ and the point $A$ has coordinates $(4,1,3)$.

   (a)

   Which of the following is a cartesian equation of $\Pi$?

   [1 mark]

   • A$2x-y+2z=2$
   • B$2x+y+2z=6$
   • C$2x-y+2z=-6$
   • D$2x-y+2z=6$

   (b)

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

   [1 mark]

   • A$7$
   • B$\frac{13}{3}$
   • C$\frac73$
   • D$\frac{19}{3}$

   (c)

   Write down an equation of $\Pi$ in the form $\mathbf{r}=\mathbf{a}+s\mathbf{b}+t\mathbf{c}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The line $l$ has equation $\left(\mathbf{r}-(\mathbf{i}+2\mathbf{j}-\mathbf{k})\right)\times(2\mathbf{i}-\mathbf{j}+3\mathbf{k})=\mathbf{0}$.

   (a)

   Which of the following is a vector equation of $l$ in the form $\mathbf{r}=\mathbf{a}+\lambda\mathbf{b}$?

   [1 mark]

   • A$\mathbf{r}=(2\mathbf{i}-\mathbf{j}+3\mathbf{k})+\lambda(\mathbf{i}+2\mathbf{j}-\mathbf{k})$
   • B$\mathbf{r}=(\mathbf{i}+2\mathbf{j}-\mathbf{k})+\lambda(2\mathbf{i}-\mathbf{j}+3\mathbf{k})$
   • C$\mathbf{r}=(-\mathbf{i}-2\mathbf{j}+\mathbf{k})+\lambda(2\mathbf{i}-\mathbf{j}+3\mathbf{k})$
   • D$\mathbf{r}=(\mathbf{i}+2\mathbf{j}-\mathbf{k})+\lambda(2\mathbf{i}-\mathbf{j}-3\mathbf{k})$

   (b)

   Which of the following is a cartesian equation of $l$?

   [1 mark]

   • A$\frac{x-1}{2}=\frac{y-2}{-1}=\frac{z+1}{3}$
   • B$\frac{x+1}{2}=\frac{y+2}{-1}=\frac{z-1}{3}$
   • C$\frac{x-1}{2}=\frac{y-2}{1}=\frac{z+1}{3}$
   • D$\frac{x-2}{1}=\frac{y+1}{2}=\frac{z-3}{-1}$

   (c)

   Find the coordinates of the point where $l$ meets the plane $y=0$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The planes $\Pi_1$ and $\Pi_2$ have equations $x+2y-z=4$ and $2x-y+3z=3$ respectively.

   (a)

   Find a vector equation of the line of intersection $l$ of $\Pi_1$ and $\Pi_2$.

   [3 marks]

   (b)

   The plane $\Pi_3$ has equation $x+y-z=6$. Find the coordinates of the point where $l$ meets $\Pi_3$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The lines $L_1$ and $L_2$ have equations $\mathbf{r}=(\mathbf{i}+2\mathbf{k})+\lambda(\mathbf{i}+2\mathbf{j}+2\mathbf{k})$ and $\mathbf{r}=(2\mathbf{i}-\mathbf{j}+4\mathbf{k})+\mu(2\mathbf{i}+\mathbf{j}-2\mathbf{k})$.

   (a)

   Show that $L_1$ and $L_2$ are skew. Hence find the equation of the plane containing $L_1$ that is parallel to $L_2$, in the form $\mathbf{r}\cdot\mathbf{n}=p$.

   [6 marks]

   (b)

   Use your answer to (a) to find the shortest distance between $L_1$ and $L_2$.

   [6 marks]

   Total for question 4: 12 marks

End of questions