Vector and Cartesian equations of planesAQA A-Level Further Maths: Subtopic test

1. 1
   The plane $\Pi$ has vector equation $\mathbf r=\begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}+\lambda\begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix}+\mu\begin{pmatrix} 0 \\ 3 \\ 1 \end{pmatrix}$.

   (a)

   Which vector is perpendicular to the plane $\Pi$?

   [1 mark]

   • A$\begin{pmatrix} 1 \\ 5 \\ 1 \end{pmatrix}$
   • B$\begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}$
   • C$\begin{pmatrix} 2 \\ 1 \\ 3 \end{pmatrix}$
   • D$\begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix}$

   (b)

   Which is a Cartesian equation of $\Pi$?

   [1 mark]

   • A$2x-y+3z=0$
   • B$x+2y=1$
   • C$2x-y+3z=8$
   • D$2x-y+3z=2$

   (c)

   Determine whether the point $(3,4,2)$ lies on $\Pi$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The points $A(1,2,0)$, $B(3,2,1)$ and $C(2,5,2)$ lie in a plane $\Pi$.

   (a)

   Which pair of vectors lies in the plane $\Pi$?

   [1 mark]

   • A$\begin{pmatrix} 2 \\ 0 \\ 1 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ 3 \\ 2 \end{pmatrix}$
   • B$\begin{pmatrix} 2 \\ 0 \\ 1 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ 3 \\ -2 \end{pmatrix}$
   • C$\begin{pmatrix} 3 \\ 2 \\ 1 \end{pmatrix}$ and $\begin{pmatrix} 2 \\ 5 \\ 2 \end{pmatrix}$
   • D$\begin{pmatrix} 2 \\ 0 \\ -1 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ 3 \\ 2 \end{pmatrix}$

   (b)

   Which is a normal vector to the plane $\Pi$?

   [1 mark]

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

   (c)

   Hence find a Cartesian equation of $\Pi$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The plane $\Pi$ has Cartesian equation $3x-2y+z=12$.

   (a)

   Find a vector equation of $\Pi$ in the form $\mathbf r=\mathbf a+\lambda\mathbf b+\mu\mathbf c$.

   [3 marks]

   (b)

   The plane $\Pi_2$ is parallel to $\Pi$ and contains the point $(1,-1,5)$. Find a Cartesian equation of $\Pi_2$, and determine whether the point $(4,1,0)$ lies on $\Pi_2$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A flat roof panel contains the points $A(2,0,3)$, $B(5,2,4)$ and $C(3,4,2)$, where $x$ and $y$ are horizontal and $z$ is vertical, with distances in metres.

   (a)

   (i) Find a vector equation of the plane containing the panel.
   (ii) Find a Cartesian equation of the plane.

   [6 marks]

   (b)

   A support cable is attached to the panel at the point $D(2,5,1)$ and runs perpendicular to the panel. Show that $D$ lies in the plane of the panel, and find a vector equation and a Cartesian equation of the line of the cable.

   [6 marks]

   Total for question 4: 12 marks

End of questions