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

1. 1
   The line $l$ passes through the points $A(1,2,-1)$ and $B(3,-2,3)$.

   (a)

   Find $\overrightarrow{AB}$.

   [1 mark]

   • A$\begin{pmatrix} 2 \\ -4 \\ 4 \end{pmatrix}$
   • B$\begin{pmatrix} 4 \\ 0 \\ 2 \end{pmatrix}$
   • C$\begin{pmatrix} -2 \\ 4 \\ -4 \end{pmatrix}$
   • D$\begin{pmatrix} 2 \\ -4 \\ -4 \end{pmatrix}$

   (b)

   Which is a vector equation of $l$?

   [1 mark]

   • A$\mathbf r=\begin{pmatrix} 2 \\ -4 \\ 4 \end{pmatrix}+\lambda\begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}$
   • B$\mathbf r=\begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}+\lambda\begin{pmatrix} 3 \\ -2 \\ 3 \end{pmatrix}$
   • C$\mathbf r=\begin{pmatrix} 3 \\ -2 \\ 3 \end{pmatrix}+\lambda\begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}$
   • D$\mathbf r=\begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}+\lambda\begin{pmatrix} 2 \\ -4 \\ 4 \end{pmatrix}$

   (c)

   Show that the point $C(5,-6,7)$ lies on $l$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A line $l_1$ has Cartesian equation $\frac{x-2}{3}=\frac{y+1}{-2}=\frac{z-4}{5}$.

   (a)

   Which is a direction vector of $l_1$?

   [1 mark]

   • A$\begin{pmatrix} 2 \\ -1 \\ 4 \end{pmatrix}$
   • B$\begin{pmatrix} -2 \\ 1 \\ -4 \end{pmatrix}$
   • C$\begin{pmatrix} 3 \\ -2 \\ 5 \end{pmatrix}$
   • D$\begin{pmatrix} 3 \\ 2 \\ 5 \end{pmatrix}$

   (b)

   Which point lies on $l_1$?

   [1 mark]

   • A$(5,1,9)$
   • B$(5,-3,9)$
   • C$(-2,1,-4)$
   • D$(3,-3,9)$

   (c)

   Find the coordinates of the point on $l_1$ at which $z=-1$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The line $l_2$ has vector equation $\mathbf r=\begin{pmatrix} 4 \\ -3 \\ 1 \end{pmatrix}+\mu\begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}$.

   (a)

   Find a Cartesian equation of $l_2$.

   [3 marks]

   (b)

   The line $l_4$ passes through the point $(1,0,5)$ and is parallel to $l_2$. Write down a vector equation of $l_4$, and show that the point $(-3,2,-1)$ lies on $l_4$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A cable is stretched in a straight line from a point $A(2,-1,5)$ to a point $B(8,2,2)$, where distances are in metres.

   (a)

   (i) Find a vector equation of the line $AB$.
   (ii) Find a Cartesian equation of the line $AB$.
   (iii) The cable is extended in a straight line beyond $B$ to a post at $C(14,5,-1)$. Show that $C$ lies on the line $AB$.

   [6 marks]

   (b)

   A second cable lies along the line with vector equation $\mathbf r=\begin{pmatrix} 1 \\ 4 \\ -2 \end{pmatrix}+\mu\begin{pmatrix} 4 \\ 2 \\ -2 \end{pmatrix}$. Show that the second cable is parallel to $AB$ but is not part of the line $AB$, and write down a Cartesian equation of the second cable.

   [6 marks]

   Total for question 4: 12 marks

End of questions