Vector equations in 2&3D (+ angle between two lines): notes and practice questions

The path of a drone, $D_{1}$, can be modelled by the vector equation $r = \begin{pmatrix} 2 \\ -1 \\ 5 \end{pmatrix} + \lambda \begin{pmatrix} -1 \\ 2 \\ 3 \end{pmatrix}$. A communication beacon is located at point B with coordinates $(k, 3, 11)$. The drone passes through the location of the beacon.

(a) Find the value of $k$.

A second drone, $D_{2}$, starts at the point $(0, 0, 1)$ and travels on a path parallel to $D_{1}$.

(b) Write down a vector equation for the path of $D_{2}$.

Consider the line $L_1$ with vector equation $\mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}$.

(a) (i) Find a Cartesian equation for the line $L_1$.

(ii) Show that the point $P(4, 8, -4)$ lies on $L_1$.

Consider a second line $L_2$ with direction vector $\begin{pmatrix} p \\ 1 \\ 3 \end{pmatrix}$. The acute angle between $L_1$ and $L_2$ is $\theta$, where $\cos\theta = \frac{1}{3}$.

(b) Find the possible values of $p$.

Let a third line, $L_3$, be defined by the vector equation $\mathbf{r} = \begin{pmatrix} 0 \\ 4 \\ 1 \end{pmatrix} + t \begin{pmatrix} q \\ 2 \\ -4 \end{pmatrix}$. The lines $L_1$ and $L_3$ intersect at a point B.

(c) Find the value of $q$ and the coordinates of B.

A drone's flight path, denoted by line $L$, is determined by the intersection of two virtual guidance planes, $\Pi_1$ and $\Pi_2$. The equations of these planes are given by:

$\Pi_1 : x + 2y - z = 4$

$\Pi_2 : 3x - y + 2z = 1$

Verify that a vector equation of $L$ is $\mathbf{r} = \begin{pmatrix} 0 \\ 3 \\ 2 \end{pmatrix} + \lambda \begin{pmatrix} 3 \\ -5 \\ -7 \end{pmatrix}$, where $\lambda \in \mathbb{R}$.

The drone's home base is located at the origin $(0,0,0)$. Find the coordinates of the point P on the flight path $L$ that is nearest to the origin.

Two spacecraft, S1 and S2, travel along straight paths, represented by the lines $L_1$ and $L_2$ respectively. The paths of the spacecraft intersect at a docking station D. A probe is located at a point P on the path of $L_2$. This is shown in the following diagram.

The direction vector of $L_1$ is $\begin{pmatrix} 2 \\ 1 \\ -2 \end{pmatrix}$. The vector $\vec{\text{DP}}$ is given by $\begin{pmatrix} k \\ 4 \\ 0 \end{pmatrix}$, where $k \ge 0$.

The acute angle between the paths $L_1$ and $L_2$ is $\theta$, where $\cos\theta = \frac{1}{3}$.

(a) Show that $2k+4 = \sqrt{k^2+16}$.

(b) Find the value of $k$.

(c) Hence, find the shortest distance from the probe at P to the path $L_1$.

The paths $L_1$ and $L_2$ lie on a plane, $\Pi$.

(d) Find a vector normal to the plane $\Pi$.

A satellite dish is modelled as a right circular cone with its vertex at V. The base of the cone lies in the plane $\Pi$ and is centred at P. The path $L_1$ is tangent to the circular base of the cone. The volume of the cone is $\frac{128\pi\sqrt{2}}{9}$ cubic units. The position vector of P is $\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$.

(e) Find the two possible position vectors for V.

Two automated guided vehicles (AGVs), AGV-1 and AGV-2, are moving along straight paths in a 3D warehouse. The path of AGV-1 is given by the vector equation $\mathbf{r}_1 = \begin{pmatrix} 1 \\ 4 \\ -1 \end{pmatrix} +s\begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}$ where $s\in \mathbb{R}$.

The path of AGV-2 is given by the vector equation $\mathbf{r}_2 = \begin{pmatrix} 5 \\ 4 \\ 2 \end{pmatrix} +t\begin{pmatrix} 3 \\ -1 \\ 1 \end{pmatrix}$ where $t\in \mathbb{R}$.

All coordinates are in meters.

(a) Show that the paths of AGV-1 and AGV-2 intersect at a point P and find the position vector of P.

(b) A safety sensor plane $\Pi$ is installed in the warehouse. The plane is given by the equation $\mathbf{r}\cdot \begin{pmatrix} 3 \\ 5 \\ -4 \end{pmatrix} =27$. Verify that the paths of both AGV-1 and AGV-2 lie entirely within this safety sensor plane $\Pi$.

(c) An emergency charging station is located at point Q with position vector $\begin{pmatrix} 1 \\ -2 \\ 4 \end{pmatrix}$.

(i) A charging drone is dispatched from Q and travels along a path perpendicular to the plane $\Pi$. This drone lands on the plane $\Pi$ at point R. Find the position vector of R.

(ii) Calculate the shortest distance from the emergency charging station Q to the safety sensor plane $\Pi$.

(d) Due to a system malfunction, AGV-1 needs to be redirected to a virtual point Q' which is the reflection of the emergency charging station Q in the plane $\Pi$. Find the position vector of Q'.

The plane $\Pi_1$ has equation $x - 2y + 3z = 1$.

(a) Show that the point $A(2, 2, 1)$ lies on the plane $\Pi_1$.

The plane $\Pi_2$ is given by $ax + (a^2 - 15)y + bz = d$, where $a, b, d \in \mathbb{R}$ and $b \neq 0$.

(b) In the case where $b = 5$, $\Pi_2$ is perpendicular to $\Pi_1$ and point A lies on $\Pi_2$. Given that $a > 0$, find the value of $a$ and the value of $d$.

For parts (c), (d) and (e) it is now given that $\Pi_2$ is parallel to $\Pi_1$.

(c) Given that $a=3$, determine the value of $b$.

It is also given that $d = 31$.

The line through A that is perpendicular to $\Pi_1$ meets $\Pi_2$ at the point B.

(d) (i) Find the coordinates of B.

(ii) Hence, find the perpendicular distance between $\Pi_1$ and $\Pi_2$.

(e) Find the equation of a third parallel plane $\Pi_3$ which is also a perpendicular distance of $\frac{2\sqrt{14}}{3}$ from $\Pi_1$.

(a) The line $L_1$ passes through the point Q(2, 0, 5) and has a direction vector $\begin{pmatrix} 1 \\ -2 \\ 2 \end{pmatrix}$.

Write down a vector equation for $L_1$.

(b) A second line, $L_2$, passes through the points C(3, 1, 0) and D(4, 3, -2).

Find a vector equation for $L_2$.

(c) Show that $L_1$ and $L_2$ are skew.

(d) Find $\vec{\text{QM}} \cdot \vec{\text{CD}}$ in terms of $\mu$, where M is a general point on $L_2$.

(e) Hence, find the coordinates of the point M on $L_2$ that is closest to Q.

(f) The origin is denoted by O(0, 0, 0). Find the equation of the plane that contains the points O, Q and the point M found in part (e). Give your answer in the form $ax + by + cz = d$, where $a, b, c, d \in \mathbb{Z}$.

A drone's initial flight path from its base station at the origin is represented by a displacement vector $\vec{d} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}$. Let $\alpha$, $\beta$, and $\gamma$ be the angles that $\vec{d}$ makes with the positive $x$-axis, $y$-axis, and $z$-axis respectively.

Show that $\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$.

If the drone's initial displacement vector is $\vec{d} = 4\mathbf{i} - 5\mathbf{j} + 2\mathbf{k}$ meters, calculate the angles $\alpha$, $\beta$, and $\gamma$ to one decimal place.

A security laser beam is emitted from the base station (origin) along a direction perpendicular to the drone's initial flight path. Show that the equation of the plane containing this laser beam can be expressed in the form $x\cos \alpha + y \cos \beta + z \cos \gamma = 0$, where $\alpha$, $\beta$, and $\gamma$ are the angles found in part (b).

Three points $D(1, 2, 0)$, $E(3, 0, -1)$ and $F(0, 1, 4)$ lie on the plane $\Pi_1$.

Find the vector $\vec{DE}$ and the vector $\vec{DF}$.

Hence find the equation of $\Pi_1$, expressing your answer in the form $ax + by + cz = d$, where $a, b, c, d \in \mathbb{Z}$.

Plane $\Pi_2$ has equation $x + y - z = 2$.

The line $L$ is the intersection of $\Pi_1$ and $\Pi_2$. Verify that the vector equation of $L$ can be written as $\mathbf{r} = \begin{pmatrix} -1 \\ 4 \\ 1 \end{pmatrix} + \lambda \begin{pmatrix} -11 \\ 13 \\ 2 \end{pmatrix}$.

The plane $\Pi_3$ is given by $x + 2y - z = 5$. The line $L$ and the plane $\Pi_3$ intersect at the point $P$.

Show that at the point $P$, $\lambda=-\frac{1}{13}$.

Hence find the coordinates of $P$.

The point $Q(-1, 4, 1)$ lies on $L$.

Find the reflection of the point $Q$ in the plane $\Pi_3$.

Hence find the vector equation of the line formed when $L$ is reflected in the plane $\Pi_3$.

A satellite dish is positioned at a ground control station $A(3, 1, 4)$. The dish is designed to track a celestial object whose path can be modelled by a line $L_1$ with vector equation $\mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix} + t \begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}$, where $t \in \mathbb{R}$.

The plane $\Pi_1$ of the satellite dish contains the line $L_1$ and passes through the ground control station $A$.

Show that the Cartesian equation of the plane $\Pi_1$ is $x + 2y = 5$.

Consider three large display screens in a museum, represented by the planes:

$\Pi_1 : x + 2y = 5$

$\Pi_2 : 3x + ay - z = 7$

$\Pi_3 : 2x - y + z = k$

where $a, k \in \mathbb{Q}$.

For a special holographic effect, the three planes must intersect along a single line.

Find the value of $a$ and the value of $k$.

Two drones, Drone X and Drone Y, have position vectors with respect to an origin O given respectively by

$\boldsymbol{r}_X = \begin{pmatrix} 10 \\ -2 \\ 2 \end{pmatrix} + t \begin{pmatrix} -4 \\ 1 \\ 3 \end{pmatrix}$

$\boldsymbol{r}_Y = \begin{pmatrix} -1 \\ -2 \\ 9 \end{pmatrix} + t \begin{pmatrix} 3 \\ 2 \\ -1 \end{pmatrix}$

where $t$ represents the time in minutes and $0 \le t \le 3$.

Entries in each column vector give the displacement east of O, the displacement north of O and the distance above sea level, all measured in kilometres.

(a) Find the three-figure bearing on which Drone Y is travelling.

(b) Show that Drone X travels at a greater speed than Drone Y.

(c) Find the acute angle between the two drones' lines of flight. Give your answer in degrees.

The two drones' lines of flight cross at point P.

(d) (i) Find the coordinates of P.

(ii) Determine the length of time between the first drone arriving at P and the second drone arriving at P.

(e) Let $D(t)$ represent the distance between Drone X and Drone Y for $0 \le t \le 3$.

Find the minimum value of $D(t)$.

A laser beam is emitted from a source at point $A(3, -1, 5)$. The beam reflects off a flat mirror which lies on the plane $\pi: x - 2y + 2z = 6$. The reflected beam appears to originate from a virtual source at point $B$, where $B$ is the reflection of $A$ in the plane $\pi$.

Determine the coordinates of $B$.

Find the exact distance between the laser source $A$ and the virtual source $B$.

Two drones, Drone Alpha and Drone Beta, are flying in a 3D space. At a particular instant, Drone Alpha passes through point $A(1, 2, 3)$ and then point $B(5, 0, 1)$.

(a) Find a vector equation of the line representing Drone Alpha's path.

At the same instant, Drone Beta passes through point $C(2, -1, 4)$ and then point $D(0, 3, 2)$.

(b) Find a vector equation of the line representing Drone Beta's path.

(c) Hence, or otherwise, find the shortest distance between the paths of Drone Alpha and Drone Beta.

A plane $\Pi$ has the Cartesian equation $x + 3y - 2z = 5$. A point B has coordinates $(1, 0, 4)$.

(a) Find the vector equation of the line $L$ that passes through the point B and is perpendicular to the plane $\Pi$.

(b) Find the coordinates of the point of intersection, N, of the line $L$ and the plane $\Pi$. Hence, find the exact distance between the point B and the plane $\Pi$.

(c) The point P has coordinates $(x_0, y_0, z_0)$.

Show that the distance between the point P and the plane $\Pi$ is given by

$\frac{|x_0 + 3y_0 - 2z_0 - 5|}{\sqrt{14}}$

A laser beam is modelled by the line $L$ with equation $\vec{r} = \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 0 \\ -2 \end{pmatrix}$. The beam strikes a flat mirror surface, which lies on the plane $\Pi$ with equation $2x - y + 3z = 5$.

(a) Find the coordinates of the point where the laser beam hits the mirror.

(b) Determine the acute angle between the laser beam and the mirror surface.

(c) Find the vector equation of the reflected laser beam.

A drone is programmed to follow a straight flight path $L$. The path is described by the Cartesian equation $L: \frac{x+3}{4} = \frac{y-1}{2} = z+2$.

Find the vector equation of the drone's flight path $L$, expressing your answer in the form $\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}$, where $\lambda \in \mathbb{R}$.

A ground control station is located at the origin $O(0,0,0)$.

Determine the minimum distance from the ground control station to the drone's flight path $L$.

A security laser grid is set up, forming a plane $\Pi$ with the equation $\Pi: x+y-6z=10$.

Verify that the drone's flight path $L$ lies entirely within the security laser grid $\Pi$.

A second drone is launched from a point $P(5, 2, -1)$. This drone's flight path, $M$, is parallel to the security laser grid $\Pi$ and is designed to intersect the $y$-axis.

Find the vector equation of the second drone's flight path $M$, expressing your answer in the form $\mathbf{s} = \mathbf{c} + \mu\mathbf{d}$, where $\mu \in \mathbb{R}$.

Let P(1, 0, 1), Q(1, 2, 0), and R(k+1, 1, -1) be three points in $\mathbb{R}^3$, where k > 0.

Let $Π$ be the plane containing the points P, Q, and R.

(a) Find a Cartesian equation for the plane $Π$ in terms of k.

(b) Let N be the midpoint of the line segment [PR]. A line L passes through N and is perpendicular to the plane $Π$. Find a vector equation for the line L in terms of k.

(c) Let $L'$ be the line defined by the equations $y=x, z=1$. Show that the line L does not intersect the line $L'$ for any k > 0.