3.11 Vector equation of a lineIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   A line $l$ has vector equation $\mathbf{r}=\begin{pmatrix}2 \\ -1 \\ 4\end{pmatrix}+\lambda\begin{pmatrix}3 \\ 2 \\ -1\end{pmatrix}$, where $\lambda\in\mathbb{R}$.

   (a)

   Which of the following points lies on $l$?

   [1 mark]

   • A$(8,\,3,\,2)$
   • B$(8,\,3,\,6)$
   • C$(5,\,1,\,5)$
   • D$(1,\,-3,\,5)$

   (b)

   Which vector is parallel to $l$?

   [1 mark]

   • A$\begin{pmatrix}3 \\ 2 \\ 1\end{pmatrix}$
   • B$\begin{pmatrix}2 \\ -1 \\ 4\end{pmatrix}$
   • C$\begin{pmatrix}6 \\ 4 \\ 2\end{pmatrix}$
   • D$\begin{pmatrix}-6 \\ -4 \\ 2\end{pmatrix}$

   (c)

   Find the coordinates of the point where $l$ meets the plane $z=0$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The line $l_1$ passes through the points $A(1,2,-3)$ and $B(4,6,9)$.

   (a)

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

   [1 mark]

   • A$\mathbf{r}=\begin{pmatrix}1 \\ 2 \\ -3\end{pmatrix}+\lambda\begin{pmatrix}4 \\ 6 \\ 9\end{pmatrix}$
   • B$\mathbf{r}=\begin{pmatrix}4 \\ 6 \\ 9\end{pmatrix}+\lambda\begin{pmatrix}3 \\ 4 \\ 12\end{pmatrix}$
   • C$\mathbf{r}=\begin{pmatrix}4 \\ 6 \\ 9\end{pmatrix}+\lambda\begin{pmatrix}5 \\ 8 \\ 6\end{pmatrix}$
   • D$\mathbf{r}=\begin{pmatrix}3 \\ 4 \\ 12\end{pmatrix}+\lambda\begin{pmatrix}1 \\ 2 \\ -3\end{pmatrix}$

   (b)

   The point $(7,\,10,\,p)$ lies on $l_1$. Find $p$.

   [1 mark]

   • A$9$
   • B$15$
   • C$21$
   • D$24$

   (c)

   Write down the parametric equations of $l_1$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A lifeboat leaves the harbour at $A(2,5)$ and travels in a straight line through the point $B(11,-7)$. Coordinates are in km, relative to a lighthouse at the origin $O$.

   (a)

   Find a vector equation of the line along which the lifeboat travels.

   [3 marks]

   (b)

   (i) Show that the point $C(20,-19)$ lies on the line.
   (ii) Find the distance $AC$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A drone's position, in metres relative to a launch point $O$, is $\mathbf{r}=\begin{pmatrix}-20 \\ 10 \\ 30\end{pmatrix}+t\begin{pmatrix}4 \\ -2 \\ -3\end{pmatrix}$, where $t\ge0$ is the time in seconds since it was first detected. The $z$-axis is vertical, and the ground is $z=0$.

   (a)

   (i) Write down the position of the drone when it is first detected.
   (ii) Find the speed of the drone.
   (iii) Find the time at which the drone lands, and the coordinates of the landing point.

   [6 marks]

   (b)

   (i) Write down the parametric equations of the path of the drone.
   (ii) Show that the drone passes directly above $O$, and find its height at that moment.
   (iii) Find the position of the drone when it is $12$ m above the ground.

   [6 marks]

   Total for question 4: 12 marks

End of questions