Planes and the scalar productEdexcel A-Level Further Maths: Subtopic test

1. 1
   Let $\mathbf p=3\mathbf i+2\mathbf j-\mathbf k$ and $\mathbf q=\mathbf i-4\mathbf j+2\mathbf k$.

   (a)

   Find $\mathbf p\cdot\mathbf q$.

   [1 mark]

   • A$-7$
   • B$9$
   • C$3\mathbf i-8\mathbf j-2\mathbf k$
   • D$13$

   (b)

   The vector $\lambda\mathbf i+\mathbf j+5\mathbf k$ is perpendicular to $\mathbf p$. Find $\lambda$.

   [1 mark]

   • A$-\frac73$
   • B$-1$
   • C$3$
   • D$1$

   (c)

   Find the angle between $\mathbf p$ and $\mathbf q$, giving your answer in degrees to 1 decimal place.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The plane $\Pi_1$ has vector equation $\mathbf r=\begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}+\lambda\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}+\mu\begin{pmatrix} 0 \\ 1 \\ 3 \end{pmatrix}$.

   (a)

   Which vector is perpendicular to the plane $\Pi_1$?

   [1 mark]

   • A$(1,-1,3)$
   • B$(3,3,1)$
   • C$(-3,3,2)$
   • D$(3,-3,1)$

   (b)

   Which is a Cartesian equation of $\Pi_1$?

   [1 mark]

   • A$3x-3y+z=3$
   • B$3x-3y+z=5$
   • C$x+y+3z=7$
   • D$3x+3y+z=5$

   (c)

   The point $(k,2,4)$ lies on $\Pi_1$. Find the value of $k$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The line $l$ has equation $\mathbf r=\begin{pmatrix} 0 \\ 1 \\ 2 \end{pmatrix}+t\begin{pmatrix} 2 \\ -1 \\ 2 \end{pmatrix}$ and the plane $\Pi$ has equation $x+2y-2z=6$.

   (a)

   Find the coordinates of the point where $l$ meets $\Pi$.

   [3 marks]

   (b)

   Find the acute angle between $l$ and $\Pi$, giving your answer in degrees to 1 decimal place.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The plane $\Pi_1$ passes through the points $A(1,0,2)$, $B(3,1,1)$ and $C(2,2,4)$. The plane $\Pi_2$ has equation $x+2y-2z=5$.

   (a)

   Find a vector equation of $\Pi_1$, and show that a Cartesian equation of $\Pi_1$ is $4x-5y+3z=10$.

   [6 marks]

   (b)

   (i) Find the acute angle between $\Pi_1$ and $\Pi_2$, giving your answer in degrees to 1 decimal place.
   (ii) The plane $\Pi_3$ has equation $x+2y+kz=4$. Given that $\Pi_3$ is perpendicular to $\Pi_1$, find the value of $k$.

   [6 marks]

   Total for question 4: 12 marks

End of questions