Unit 21 Review Exercises
Vectors II
These questions mix the ideas from the whole unit. All of them are HL content. Paper 1 means no calculator; Paper 2 means a GDC is allowed.
1 Paper 1 AHL The line \(l\) passes through \(A(1, -2, 4)\) and \(B(3, 1, 0)\).
(a) Find a vector equation of \(l\), and its Cartesian equations.
(b) Determine whether the point \(C(7, 7, -8)\) lies on \(l\).
(a) \(\overrightarrow{AB} = \begin{pmatrix} 2 \\ 3 \\ -4 \end{pmatrix}\), so \(\boldsymbol{r} = \begin{pmatrix} 1 \\ -2 \\ 4 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 3 \\ -4 \end{pmatrix}\), and \(\dfrac{x - 1}{2} = \dfrac{y + 2}{3} = \dfrac{z - 4}{-4}\).
(b) \(1 + 2\lambda = 7 \Rightarrow \lambda = 3\); \(y = -2 + 9 = 7\) ✓; \(z = 4 - 12 = -8\) ✓. Yes, \(C\) lies on \(l\). See 21.1 and 21.2.
2 Paper 1 AHL A line has Cartesian equations \(\dfrac{4 - x}{2} = \dfrac{y + 1}{3}\), \(\;z = 5\).
(a) Write the line in vector form.
(b) Find the point on the line where \(y = 8\).
(a) \(\dfrac{4 - x}{2} = \dfrac{x - 4}{-2}\), and \(z\) is constant, so \(\boldsymbol{r} = \begin{pmatrix} 4 \\ -1 \\ 5 \end{pmatrix} + \lambda \begin{pmatrix} -2 \\ 3 \\ 0 \end{pmatrix}\).
(b) \(-1 + 3\lambda = 8 \Rightarrow \lambda = 3\), giving \((-2, 8, 5)\). See 21.1.
3 Paper 1 AHL The line \(l\) is \(\boldsymbol{r} = \begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}\) and \(P\) is the point \((5, 3, 0)\).
(a) Find the foot of the perpendicular from \(P\) to \(l\).
(b) Find the shortest distance from \(P\) to \(l\).
(c) Find the reflection of \(P\) in \(l\).
(a) \(\overrightarrow{PF} = \begin{pmatrix} \lambda - 4 \\ -1 \\ \lambda \end{pmatrix}\); \(\;\overrightarrow{PF} \cdot \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} = 2\lambda - 4 = 0\), so \(\lambda = 2\) and \(F = (3, 2, 2)\).
(b) \(\overrightarrow{PF} = \begin{pmatrix} -2 \\ -1 \\ 2 \end{pmatrix}\), distance \(= 3\).
(c) \(\overrightarrow{OP'} = \overrightarrow{OP} + 2\overrightarrow{PF} = \begin{pmatrix} 1 \\ 1 \\ 4 \end{pmatrix}\), so \(P' = (1, 1, 4)\). See 21.2.
4 Paper 2 AHL The lines \(l_1: \boldsymbol{r} = \begin{pmatrix} 3 \\ -1 \\ 2 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}\) and \(l_2: \boldsymbol{r} = \begin{pmatrix} 0 \\ 3 \\ -5 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}\) meet at the point \(X\).
(a) Find the coordinates of \(X\).
(b) Find the acute angle between the lines.
(a) \(3 + \lambda = 2\mu\), \(\;-1 + 2\lambda = 3 - \mu\). From the second, \(\mu = 4 - 2\lambda\); then \(3 + \lambda = 8 - 4\lambda \Rightarrow \lambda = 1\), \(\mu = 2\). Check \(z\): \(2 - 1 = 1\) and \(-5 + 6 = 1\) ✓. So \(X = (4, 1, 1)\).
(b) \(\boldsymbol{b}_1 \cdot \boldsymbol{b}_2 = 2 - 2 - 3 = -3\); \(\;\cos\theta = \dfrac{3}{\sqrt{6}\sqrt{14}} = \dfrac{3}{\sqrt{84}}\), so \(\theta \approx 70.9^\circ\). See 21.3.
5 Paper 1 AHL The lines \(l_1: \boldsymbol{r} = \begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 1 \\ 3 \end{pmatrix}\) and \(l_2: \boldsymbol{r} = \begin{pmatrix} 4 \\ 0 \\ k \end{pmatrix} + \mu \begin{pmatrix} 1 \\ -1 \\ 1 \end{pmatrix}\) intersect.
(a) Find the value of \(k\).
(b) Describe the relationship between the lines when \(k\) takes any other value. Give a reason.
(a) \(x\): \(1 + 2\lambda = 4 + \mu\); \(\;y\): \(\lambda = -\mu\). So \(1 - 2\mu = 4 + \mu \Rightarrow \mu = -1\), \(\lambda = 1\). \(z\): \(-1 + 3 = k - 1\), so \(k = 3\).
(b) The directions are not multiples of each other (compare \(\frac{2}{1}\) and \(\frac{1}{-1}\)), so the lines are never parallel. If \(k \ne 3\) the \(z\)-equation fails, so the lines do not meet. They are skew. See 21.3.
6 Paper 1 AHL \(l_1: \boldsymbol{r} = \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix}\) and \(l_2: \boldsymbol{r} = \begin{pmatrix} 4 \\ 4 \\ 0 \end{pmatrix} + \mu \begin{pmatrix} -2 \\ -4 \\ -4 \end{pmatrix}\).
(a) Show that the lines are parallel but not coincident.
(b) Find the distance between the two lines.
(a) \(\begin{pmatrix} -2 \\ -4 \\ -4 \end{pmatrix} = -2\begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix}\), so they are parallel. Test \((4, 4, 0)\) on \(l_1\): \(1 + \lambda = 4 \Rightarrow \lambda = 3\), but then \(y = 7 \ne 4\). Not coincident.
(b) The distance between parallel lines is the distance from any point of \(l_2\) to \(l_1\). With \(P(4, 4, 0)\): \(\overrightarrow{PF} = \begin{pmatrix} \lambda - 3 \\ 2\lambda - 3 \\ 2\lambda \end{pmatrix}\), \(\;\overrightarrow{PF} \cdot \begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix} = 9\lambda - 9 = 0\), so \(\lambda = 1\). \(\overrightarrow{PF} = \begin{pmatrix} -2 \\ -1 \\ 2 \end{pmatrix}\), and the distance is \(3\). See 21.2 and 21.3.
7 Paper 2 AHL Find the acute angle between the line through \(A(1, 2, 3)\) and \(B(4, -1, 5)\) and the line \(\dfrac{x - 2}{1} = \dfrac{y}{2} = \dfrac{z + 1}{-2}\).
\(\overrightarrow{AB} = \begin{pmatrix} 3 \\ -3 \\ 2 \end{pmatrix}\) and \(\boldsymbol{b} = \begin{pmatrix} 1 \\ 2 \\ -2 \end{pmatrix}\). \(\;\overrightarrow{AB} \cdot \boldsymbol{b} = 3 - 6 - 4 = -7\).
\(\cos\theta = \dfrac{7}{\sqrt{22} \times 3}\), so \(\theta \approx 60.2^\circ\). See 21.3.
8 Paper 2 AHL Two boats move with constant velocities. At time \(t\) hours their positions, in km, are \(\boldsymbol{r}_P = \begin{pmatrix} 0 \\ -5 \end{pmatrix} + t\begin{pmatrix} 3 \\ 4 \end{pmatrix}\) and \(\boldsymbol{r}_Q = \begin{pmatrix} 7 \\ 0 \end{pmatrix} + t\begin{pmatrix} -1 \\ 3 \end{pmatrix}\).
(a) Find the speed of boat \(P\).
(b) Find the time when the boats are closest together, and the minimum distance between them.
(a) \(\sqrt{9 + 16} = 5\) km h⁻¹.
(b) \(\boldsymbol{r}_Q - \boldsymbol{r}_P = \begin{pmatrix} 7 - 4t \\ 5 - t \end{pmatrix}\), so \(d^2 = (7 - 4t)^2 + (5 - t)^2 = 17t^2 - 66t + 74\).
Minimum at \(t = \dfrac{66}{34} = \dfrac{33}{17} \approx 1.94\) hours. Then \(d = \sqrt{74 - \dfrac{66^2}{68}} \approx 3.15\) km. See 21.3.
9 Paper 1 AHL The line \(l\) has direction \(\begin{pmatrix} 2 \\ -1 \\ 2 \end{pmatrix}\). The points \(A\) and \(B\) are \((1, 1, 2)\) and \((3, 5, 6)\).
(a) Find the length of projection of \(\overrightarrow{AB}\) onto \(l\).
(b) A second line has direction \(\begin{pmatrix} p \\ 1 \\ -1 \end{pmatrix}\) and is perpendicular to \(l\). Find \(p\).
(a) \(\overrightarrow{AB} = \begin{pmatrix} 2 \\ 4 \\ 4 \end{pmatrix}\). Length \(= \dfrac{|4 - 4 + 8|}{3} = \dfrac{8}{3}\).
(b) \(2p - 1 - 2 = 0 \Rightarrow p = \dfrac{3}{2}\). See 21.3.
10 Paper 1 AHL The line \(l\) passes through \((0, 2, -1)\) and is parallel to the \(x\)-axis.
(a) Write down a vector equation of \(l\).
(b) Find the shortest distance from the origin \(O\) to \(l\).
(c) Find the two points on \(l\) whose distance from \(O\) is \(3\).
(a) \(\boldsymbol{r} = \begin{pmatrix} 0 \\ 2 \\ -1 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}\).
(b) A general point is \((\lambda, 2, -1)\), at distance \(\sqrt{\lambda^2 + 5}\) from \(O\). The minimum is at \(\lambda = 0\): the shortest distance is \(\sqrt{5}\).
(c) \(\lambda^2 + 5 = 9 \Rightarrow \lambda = \pm 2\): the points \((2, 2, -1)\) and \((-2, 2, -1)\). See 21.1 and 21.2.