Angles between planes and between lines and planesAQA A-Level Further Maths: Subtopic test

1. 1
   Plane $\Pi_1$ has equation $2x+y-2z=5$ and plane $\Pi_2$ has equation $2x+3y+6z=11$.

   (a)

   Find the scalar product $\mathbf n_1\cdot\mathbf n_2$ of the normal vectors $\mathbf n_1=(2,1,-2)$ and $\mathbf n_2=(2,3,6)$ of the two planes.

   [1 mark]

   • A$5$
   • B$19$
   • C$-5$
   • D$21$

   (b)

   Find the acute angle between $\Pi_1$ and $\Pi_2$.

   [1 mark]

   • A$76.2^\circ$
   • B$103.8^\circ$
   • C$13.8^\circ$
   • D$60^\circ$

   (c)

   A third plane $\Pi_3$ has equation $x+ky+2z=1$. Given that $\Pi_1$ and $\Pi_3$ are perpendicular, find the value of $k$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The line $l$ has equation $\mathbf r=\begin{pmatrix}1 \\ -1 \\ 3\end{pmatrix}+\lambda\begin{pmatrix}2 \\ -1 \\ 2\end{pmatrix}$ and the plane $\Pi$ has equation $x+2y+2z=9$.

   (a)

   Find the scalar product of the direction vector of $l$ and the normal vector of $\Pi$.

   [1 mark]

   • A$8$
   • B$9$
   • C$0$
   • D$4$

   (b)

   A second line $m$ has direction vector $2\mathbf i-\mathbf j$. Which statement about $m$ and $\Pi$ is correct?

   [1 mark]

   • A$m$ is perpendicular to $\Pi$
   • B$m$ is parallel to $\Pi$ (or lies in it)
   • C$m$ makes an angle of $45^\circ$ with $\Pi$
   • D$m$ meets $\Pi$ at exactly one point

   (c)

   Find the acute angle between $l$ and $\Pi$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Plane $\Pi_1$ has equation $x+2y+2z=3$ and plane $\Pi_2$ has equation $4x-3y=7$.

   (a)

   Find the acute angle between $\Pi_1$ and $\Pi_2$.

   [3 marks]

   (b)

   The line $L$ has equation $\mathbf r=\begin{pmatrix}1 \\ 0 \\ 5\end{pmatrix}+t\begin{pmatrix}1 \\ -2 \\ 2\end{pmatrix}$. Find the acute angle between $L$ and $\Pi_1$, and the acute angle between $L$ and $\Pi_2$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A cuboid has one vertex at the origin $O$ and its edges along the coordinate axes, with $A=(4,0,0)$, $C=(0,3,0)$ and $D=(0,0,2)$. The vertex opposite $O$ is $G=(4,3,2)$. The base of the cuboid is the plane $z=0$.

   (a)

   (i) Show that $A$, $C$ and $D$ lie on the plane $3x+4y+6z=12$, and explain why this is the equation of plane $ACD$.
   (ii) Find the acute angle between plane $ACD$ and the base of the cuboid.

   [6 marks]

   (b)

   Find the acute angle between the line $OG$ and plane $ACD$, and the acute angle between $OG$ and the base of the cuboid.

   [6 marks]

   Total for question 4: 12 marks

End of questions