Vector equations of linesEdexcel International A Level Maths: Subtopic test

1. 1
   The line $l_1$ passes through the points $A(2,-1,3)$ and $B(5,1,1)$.

   (a)

   Which of the following is a vector equation of the line $l_1$?

   [1 mark]

   • A$\mathbf{r}=(2\mathbf{i}-\mathbf{j}+3\mathbf{k})+t(3\mathbf{i}+2\mathbf{j}-2\mathbf{k})$
   • B$\mathbf{r}=(2\mathbf{i}-\mathbf{j}+3\mathbf{k})+t(7\mathbf{i}+0\mathbf{j}+4\mathbf{k})$
   • C$\mathbf{r}=(5\mathbf{i}+\mathbf{j}+\mathbf{k})+t(2\mathbf{i}-\mathbf{j}+3\mathbf{k})$
   • D$\mathbf{r}=(3\mathbf{i}+2\mathbf{j}-2\mathbf{k})+t(2\mathbf{i}-\mathbf{j}+3\mathbf{k})$

   (b)

   Which of the following points lies on $l_1$?

   [1 mark]

   • A$(8,3,1)$
   • B$(5,3,-1)$
   • C$(7,1,-1)$
   • D$(8,3,-1)$

   (c)

   Find the coordinates of the point where $l_1$ crosses the plane $z=0$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The line $m$ passes through the points $C(1,4,-2)$ and $D(3,3,0)$.

   (a)

   Which of the following is the direction vector of $m$?

   [1 mark]

   • A$4\mathbf{i}+7\mathbf{j}-2\mathbf{k}$
   • B$2\mathbf{i}+\mathbf{j}+2\mathbf{k}$
   • C$2\mathbf{i}-\mathbf{j}+2\mathbf{k}$
   • D$3\mathbf{i}+3\mathbf{j}$

   (b)

   The line $m$ has equation $\mathbf{r}=\mathbf{c}+t(\mathbf{d}-\mathbf{c})$. Which of the following points lies on $m$?

   [1 mark]

   • A$(7,1,2)$
   • B$(7,1,4)$
   • C$(7,2,4)$
   • D$(6,1,4)$

   (c)

   Determine whether the point $E(-3,6,-6)$ lies on $m$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Line $l_1$ has equation $\mathbf{r}=(\mathbf{i}+\mathbf{j}+2\mathbf{k})+s(2\mathbf{i}+\mathbf{j}-\mathbf{k})$ and line $l_2$ has equation $\mathbf{r}=(\mathbf{i}+4\mathbf{j}+5\mathbf{k})+t(\mathbf{i}-\mathbf{j}-2\mathbf{k})$, where $s$ and $t$ are scalar parameters.

   (a)

   Show that $l_1$ and $l_2$ intersect, and find the coordinates of the point of intersection.

   [3 marks]

   (b)

   The point $A$ is on $l_1$ when $s=0$ and the point $B$ is on $l_2$ when $t=0$. Find a vector equation of the line $l_3$ through $A$ and $B$, and the distance $AB$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Two drones fly in straight lines. Relative to a control tower at the origin $O$, with distances in km and $s$ and $t$ in minutes, drone $P$ has position $\mathbf{r}=(2\mathbf{i}+\mathbf{j})+s(\mathbf{i}+2\mathbf{j}+3\mathbf{k})$ and drone $Q$ has position $\mathbf{r}=(-\mathbf{i}+4\mathbf{k})+t(2\mathbf{i}-\mathbf{j}+\mathbf{k})$. The flight path of $P$ is the line $l_3$ and the flight path of $Q$ is the line $l_4$.

   (a)

   Show that the flight paths $l_3$ and $l_4$ are skew lines.

   [6 marks]

   (b)

   Find the speed of drone $P$, and its distance from the control tower after 4 minutes. The drone is required to stay within 15 km of the tower; state whether it does.

   [6 marks]

   Total for question 4: 12 marks

End of questions