Vector basics (position, displacement vectors, components, ijk, vector algebra, magnitude): notes and practice questions

A drone is programmed to fly between three waypoints in a 3D space. The coordinates of the waypoints are given as P1(3, 7, 2), P2(8, 4, 10), and P3(1, 9, 5).

(a) Calculate the distance the drone travels from waypoint P1 to waypoint P2.

(b) Calculate the angle formed by the drone's path at waypoint P2 (i.e., the angle P1P2P3). Give your answer in radians to three significant figures.

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.

A geological survey is being conducted in a mountainous region. Three sensor stations, A, B, and C, are set up at different locations. Their coordinates, relative to a central reference point (in meters), are given as:

Station A: $(1, 2, 3)$

Station B: $(4, 5, 6)$

Station C: $(7, 2, 1)$

(a) Find the distance between Station A and Station B.

(b) Find the size of the angle $\text{ABC}$ (the angle at Station B).

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 $\Pi_1$.

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 diagram shows a triangle OXY with $\vec{OX} = \mathbf{x}$ and $\vec{OY} = \mathbf{y}$.

The point M is the midpoint of OY. The point Q lies on the line segment (XY) such that $\vec{XQ} = k\vec{XY}$, where $0 < k < 1$.

(a) Show that $\vec{MQ} = (1-k)\mathbf{x} + (k - \frac{1}{2})\mathbf{y}$.

(b) It is given that $|\mathbf{x}| = 3$, $|\mathbf{y}| = 4$ and the angle between vectors $\mathbf{x}$ and $\mathbf{y}$ is $60^\circ$.

In the case that $\vec{MQ}$ is perpendicular to $\vec{XY}$, find the value of $k$.

A landscape architect is designing a triangular shade sail for a patio. The vertices of the sail are defined by points A, B, and C in a 3D coordinate system, where the z-axis represents height.

The coordinates of the vertices are A$(0, k, 1)$, B$(2, 1, 0)$ and C$(k, 0, 3)$, where $k$ is a positive constant representing a design parameter.

(a) Show that the vector product $\vec{AB} \times \vec{AC}$ is given by $\begin{pmatrix} 2-3k \\ -k-4 \\ k^2-3k \end{pmatrix}$.

(b) The architect wants to minimize the tension in the sail, which is proportional to the magnitude of the vector product of two adjacent sides. Find the smallest possible value of $|\vec{AB} \times \vec{AC}|$.

(c) Calculate the smallest possible area of the shade sail.

(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 $a, b, c, d \in \mathbb{Z}$.

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 $\Pi_3$.

Points P and Q have position vectors $\vec{p}$ and $\vec{q}$ respectively, relative to an origin O. Let M be the midpoint of the line segment [PQ].

Show that the position vector of M is $\vec{m} = \frac{1}{2}(\vec{p} + \vec{q})$.

A triangle has vertices P(1, 0, 2), Q(3, 4, -2), and R(5, 2, 6).

Let L, M and N be the midpoints of the sides [PQ], [QR] and [RP] respectively. Find the position vectors of L, M and N.

The centroid G of the triangle PQR has position vector $\vec{g} = \frac{1}{3}(\vec{p} + \vec{q} + \vec{r})$.

Find the coordinates of G.

Show that the points P, G, and M are collinear, where M is the midpoint of [QR].

Hence, find the ratio PG:GM.

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 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).

(a) A deep-sea submersible is navigating a complex underwater current system. Its primary thruster provides a force vector $\mathbf{F_A} = \begin{pmatrix} 9 \\ -12 \end{pmatrix}$ N. It is also affected by a secondary current, which exerts a force $\mathbf{F_B}$ with a magnitude of 10 N.

Find the possible range of values for the magnitude of the resultant force $|\mathbf{F_A} + \mathbf{F_B}|$.

(b) Given that the magnitude of the resultant force $|\mathbf{F_A} + \mathbf{F_B}|$ is a minimum, find the resultant force vector.

(c) A third, unknown current exerts a force $\mathbf{F_C} = \begin{pmatrix} x \\ y \end{pmatrix}$ N, where $x, y \in \mathbb{R}^+$. Find $\mathbf{F_C}$ such that its magnitude is equal to the magnitude of $\mathbf{F_B}$ and it acts perpendicularly to $\mathbf{F_A}$.

Consider the points given by the coordinates $P(2, 1, 0)$, $Q(0, 3, 1)$, $R(1, 0, 4)$.

Find the vector $\vec{PQ} \times \vec{PR}$.

Hence, find the exact area of triangle PQR.

Show that the Cartesian equation of the plane $\Pi_1$, which contains the triangle PQR, is $9x + 7y + 4z = 25$.

A second plane is given by the equation $\Pi_2: x + y - z = 1$. Find a vector equation for the line of intersection of the planes $\Pi_1$ and $\Pi_2$.

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

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

Consider the non-zero vectors $\vec{u}$ and $\vec{v}$. Let $\theta$ be the angle between $\vec{u}$ and $\vec{v}$.

Using the definitions of $\vec{u} \cdot \vec{v}$ and $\vec{u} \times \vec{v}$ in terms of $|\vec{u}|$, $|\vec{v}|$ and $\theta$, show that $(\vec{u} \cdot \vec{v})^2 + |\vec{u} \times \vec{v}|^2 = |\vec{u}|^2|\vec{v}|^2$.

A triangle PQR has vertices P(1, 0, 1), Q($a, b$, 2) and R(4, 1, 1), where $a, b \in \mathbb{Q}$.

The vectors $\vec{u}$ and $\vec{v}$ are defined as $\vec{u} = \vec{PQ}$ and $\vec{v} = \vec{PR}$.

It is given that $\vec{u} \cdot \vec{v} = 4$ and the area of triangle PQR is $\frac{\sqrt{14}}{2}$ square units.

Find the value of $|\vec{u} \times \vec{v}|$.

Hence, or otherwise, find the value of $|\vec{u}|$.

Hence, or otherwise, find the possible values of $a$ and the corresponding values of $b$.

Consider a new point S, the vector $\vec{w}$ is defined as $\vec{w} = \vec{RS}$.

It is given that $\vec{u} \cdot \vec{w} = 0$ and $\vec{v} \cdot \vec{w} = 0$, and the area of triangle PRS is 10 square units.

Assuming that $a = 2$, find the possible vectors for $\vec{w}$.