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

1. 1
   The line $l_1$ passes through the points $A(2,-1,4)$ and $B(5,1,0)$.

   (a)

   Which is a vector equation of $l_1$?

   [1 mark]

   • A$\mathbf r=(2,-1,4)+\lambda(5,1,0)$
   • B$\mathbf r=(2,-1,4)+\lambda(7,0,4)$
   • C$\mathbf r=(2,-1,4)+\lambda(3,2,-4)$
   • D$\mathbf r=(3,2,-4)+\lambda(2,-1,4)$

   (b)

   Which is a Cartesian equation of $l_1$?

   [1 mark]

   • A$\frac{x+2}{3}=\frac{y-1}{2}=\frac{z+4}{-4}$
   • B$\frac{x-2}{3}=\frac{y+1}{2}=\frac{z-4}{-4}$
   • C$\frac{x-2}{3}=\frac{y+1}{2}=\frac{z-4}{4}$
   • D$\frac{x-3}{2}=\frac{y-2}{-1}=\frac{z+4}{4}$

   (c)

   Determine whether the point $C(11,5,-12)$ lies on $l_1$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The lines $l_2$ and $l_3$ have vector equations $\mathbf r=\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}+\mu\begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}$ and $\mathbf r=\begin{pmatrix} 5 \\ 0 \\ 4 \end{pmatrix}+t\begin{pmatrix} -4 \\ 2 \\ -2 \end{pmatrix}$.

   (a)

   Which statement describes the relationship between $l_2$ and $l_3$?

   [1 mark]

   • AThey intersect at a single point
   • BThey are skew
   • CThey are the same line
   • DThey are parallel but not the same line

   (b)

   Which is a Cartesian equation of $l_3$?

   [1 mark]

   • A$\frac{x-5}{-4}=\frac{y}{2}=\frac{z-4}{-2}$
   • B$\frac{x-5}{4}=\frac{y}{2}=\frac{z-4}{2}$
   • C$\frac{x+5}{-4}=\frac{y}{2}=\frac{z+4}{-2}$
   • D$\frac{x-5}{-2}=\frac{y}{-4}=\frac{z-4}{2}$

   (c)

   Explain why $l_2$ and $l_3$ do not intersect.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Three lines are given by $l_4$: $\mathbf r=\begin{pmatrix} 3 \\ 1 \\ -2 \end{pmatrix}+\lambda\begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}$, $l_5$: $\mathbf r=\begin{pmatrix} 3 \\ 6 \\ -7 \end{pmatrix}+\mu\begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}$ and $l_6$: $\mathbf r=\begin{pmatrix} 1 \\ 4 \\ 0 \end{pmatrix}+\nu\begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}$.

   (a)

   Show that $l_4$ and $l_5$ intersect and find the position vector of their point of intersection.

   [3 marks]

   (b)

   Show that $l_4$ and $l_6$ are skew.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Distances are in kilometres and time is in minutes. For $t\ge0$, drone $A$ has position vector $\begin{pmatrix} 2 \\ -3 \\ 5 \end{pmatrix}+t\begin{pmatrix} 1 \\ 2 \\ -2 \end{pmatrix}$ and drone $B$ has position vector $\begin{pmatrix} 10 \\ -8 \\ 4 \end{pmatrix}+t\begin{pmatrix} -2 \\ 3 \\ -1 \end{pmatrix}$. The ground is the plane $z=0$.

   (a)

   (i) Write down a Cartesian equation of the flight path of drone $A$.
   (ii) Find the coordinates of the point where drone $A$ would reach the ground.
   (iii) Find the distance drone $A$ travels from $t=0$ until it reaches the ground.

   [6 marks]

   (b)

   Show that the flight paths of $A$ and $B$ intersect, and determine whether the drones collide.

   [6 marks]

   Total for question 4: 12 marks

End of questions