Scalar product of two vectors (+ angle between two vectors): notes and practice questions

The paths of two automated guided vehicles (AGVs), $L_1$ and $L_2$, in a large warehouse are modelled by the following vector equations, where $k\in\mathbb{R}$ is a constant:

$L_1:\boldsymbol{r}=\begin{pmatrix} 1 \ 2 \ 3 \end{pmatrix} + \lambda \begin{pmatrix} k \ 5 \ -6 \end{pmatrix}$

$L_2: \boldsymbol{r} = \begin{pmatrix} 10 \ -1 \ 8 \end{pmatrix} + \mu \begin{pmatrix} k+2 \ 3 \ 5 \end{pmatrix}$

It is known that the paths of the AGVs are perpendicular.

(a) Find the possible value(s) for $k$.

(b) In the case that $k < 0$, determine whether the lines intersect.

The position of a drone, D$_1$, $t$ seconds after leaving a control tower T, is given by

$\mathbf{r} = \begin{pmatrix} 3 \\ 5 \\ 1 \end{pmatrix} + t \begin{pmatrix} -2 \\ 3 \\ 4 \end{pmatrix}$, $t \ge 0$.

The units of distance are metres.

Write down the coordinates of the control tower T.

Four seconds after leaving T, D$_1$ is at point P.

Find the displacement vector $\vec{\text{TP}}$.

Find the distance $|\vec{\text{TP}}|$.

A second drone, D$_2$, leaves the control tower T at the same time as D$_1$. D$_2$ is moving in the direction of the vector $\begin{pmatrix} 1 \\ -1 \\ 2 \end{pmatrix}$.

Find the angle between the initial flight paths of Drone 1 and Drone 2.

The drone D$_2$ has a speed of $10$ m s$^{-1}$.

Find the distance between Drone 1 and Drone 2 when $t = 4$ seconds.

A surveillance drone D is flying with a constant velocity, $\boldsymbol{v}$, measured in kilometres per hour, where

$\boldsymbol{v} = \begin{pmatrix} 8 \\ -6 \end{pmatrix}$.

At time $t = 0$ the drone is at a point A(50, 20) relative to an origin O, where distances are

measured in kilometres.

Find the position vector $\vec{OD}$ of the drone at time $t$ hours.

A protected bird's nest is located at a point N(99, 2).

Find the value of $t$ when the drone will be closest to the bird's nest.

An environmental sensor will trigger if the drone flies within 18 kilometres of the nest.

State whether the sensor will trigger. Give a reason for your answer.

(a) A ground crew on Luna Prime is tracking a supply drone. From the Aether station's control tower, the drone is observed to be $8500$ km East and $4000$ km North in a localized flat-map approximation.

(i) Find the straight-line distance from the Aether station to the drone.

(ii) Find the bearing of the drone from the Aether station. Give your answer in degrees, correct to one decimal place.

(b) The Aether station (A) is located at the origin $(0,0,0)$ of a 3D Cartesian coordinate system for this part. A research probe (P) is located at $(4000, 0, 0)$ km. A navigation beacon (B) is located at $(0, 4000, 0)$ km.

(i) Show that the position vector of the research probe, $\vec{p}$, is perpendicular to the position vector of the navigation beacon, $\vec{b}$.

(ii) Assuming the probe and beacon are on the surface of Luna Prime with a radius of $4000$ km, calculate the shortest distance along the surface between the research probe and the navigation beacon.

(c) Consider the Aether station (A) at $(4000, 0, 0)$ km, the Boreas station (B) at $(0, 4000, 0)$ km, and a North Pole reference point (N) at $(0, 0, 4000)$ km. Let $\vec{a}$, $\vec{b}$, and $\vec{n}$ be their respective position vectors from the centre of Luna Prime.

(i) Find the vector $\vec{a} \times \vec{b}$.

(ii) Show that the angle at vertex A in the spherical triangle formed by Aether, Boreas, and the North Pole (ABN) is $90^{\circ}$.

(d) A supply route between Aether and a new outpost, Delta, has an arc length of $2000$ km. Given that the radius of Luna Prime is $4000$ km, show that the central angle $\theta$ between Aether and Delta is $28.6^{\circ}$, correct to three significant figures.

(e) The Aether station (A) is located at $40^{\circ}$ N, $20^{\circ}$ E, and the Boreas station (B) is located at $70^{\circ}$ N, $100^{\circ}$ E on Luna Prime, which has a radius of $6000$ km. Find the shortest distance along the surface from Aether to Boreas.

(f) Using the vector method from part (c), find the initial bearing from Aether to Boreas. Give your answer in degrees, correct to one decimal place.

The flight path of drone A can be modelled by the vector equation $\boldsymbol{r}_A = \begin{pmatrix} 1 \\ 5 \\ -2 \end{pmatrix} + s \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}$, where $s \in \mathbb{R}$ represents time in minutes.

The flight path of drone B can be modelled by the vector equation $\boldsymbol{r}_B = \begin{pmatrix} 1 \\ 0 \\ 3 \end{pmatrix} + t \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}$, where $t \in \mathbb{R}$ represents time in minutes.

The two drones intersect at a point P. Find the coordinates of P.

Find the acute angle between the flight paths of the two drones.

A triangular section of a modern architectural roof is being designed. The vertices of this section are represented by points P, Q, and R in a 3D coordinate system, where distances are measured in metres.

Point P is at $(0, 0, 2)$.

Point Q is at $(4, 7, 3)$.

Point R is at $(9, 2, 2)$.

Calculate the vector product $\vec{PQ} \times \vec{PR}$.

Hence, find the area of the triangular roof section.

A support beam is to be installed from point Q perpendicular to the edge PR. Find the length of this support beam.

The building's base lies on the horizontal plane (z=0). Find the acute angle that the roof section makes with the horizontal plane.

(a) A drone is programmed to fly along a straight path $L$. Its position at time $t$ (in minutes) can be modelled by the vector equation $\mathbf{r} = \begin{pmatrix} 1 \\ 3 \\ 0 \end{pmatrix} + t \begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}$, where the coordinates are in metres. A stationary sensor $S$ is located at the point $(10, 4, 4)$.

Find the coordinates of the point $P$ on the drone's path that is closest to the sensor $S$.

(b) Find a vector that is perpendicular to both the drone's path $L$ and the line segment connecting point $P$ to the sensor $S$.

A drone, 'SkyRanger 1', is flying along a path that can be modelled by the line $L_1$ with equation $\mathbf{r} = \begin{pmatrix} 2 \\ 5 \\ -1 \end{pmatrix} + t \begin{pmatrix} 3 \\ -2 \\ 4 \end{pmatrix}$, where $t$ is a time parameter in minutes. A ground control station detects SkyRanger 1 at a specific point A with coordinates $(-7, 11, k)$.

(a) Calculate the value of $k$.

A second drone, 'AeroScout 2', is dispatched. Its flight path is modelled by the line $L_2$ with equation $\mathbf{r} = \begin{pmatrix} -15 \\ 7 \\ u \end{pmatrix} + s \begin{pmatrix} 2 \\ p \\ q \end{pmatrix}$, where $s$ is a time parameter in minutes. AeroScout 2 is designed to intercept SkyRanger 1 at point A and its path $L_2$ is perpendicular to the path of SkyRanger 1.

(b) Determine the values of $p$, $q$, and $u$.

A rectangular prism (cuboid) is used as a building block. It has one vertex at the origin O(0,0,0). Its dimensions are 6 units along the x-axis, 4 units along the y-axis, and 3 units along the z-axis. The vertices are given by O(0,0,0), A(6,0,0), B(6,4,0), C(0,4,0), D(0,0,3), E(6,0,3), F(6,4,3), and G(0,4,3).

Find the surface area of the cuboid.

Find the length of the space diagonal [OF].

The space diagonals [OF] and [CE] intersect at the point M.

Find the coordinates of M.

Find the acute angle between the diagonals [OF] and [CE] at their intersection point M.

(a) Two forces, $F_1 = 12$ N and $F_2 = 7$ N, are acting on an object. The angle between the directions of the two forces is $60^\circ$. Calculate the magnitude of the resultant force.

(b) If the direction of the second force, $F_2$, is reversed, determine the new magnitude of the resultant force.

A drone is programmed to fly from point $A(1, 2)$ to point $B(5, 5)$. During its flight, it experiences a constant wind force $\mathbf{W} = -10\mathbf{i} + 5\mathbf{j}$ Newtons.

Calculate the scalar component of the wind force $\mathbf{W}$ in the direction of the drone's intended path from $A$ to $B$.

Determine the scalar component of the wind force $\mathbf{W}$ which is perpendicular to the drone's intended path.

The wind force $\mathbf{W}$ can be resolved into two vector components: one acting in the direction of the drone's intended path and the other acting perpendicular to it.

State the component of the wind force $\mathbf{W}$ in the direction of the drone's intended path (from part (a) ) in vector form. Give your answer in terms of the unit vectors $\mathbf{i}$ and $\mathbf{j}$.

A robot arm applies a force $\mathbf{F}$ to move a component. The force vector is $\mathbf{F} = \begin{pmatrix} 7 \\ -3 \end{pmatrix}$ newtons (N). The component is moved from an initial position $P_1(2, 5)$ to a final position $P_2(8, 1)$, where coordinates are in metres. Calculate the work done by the robot arm.

In physics, the work done $W$ by a constant force $\mathbf{F}$ moving an object through a displacement $\vec{d}$ is generally given by the scalar product $W = \mathbf{F} \cdot \vec{d}$. Explain why, if the force is applied exactly in the direction of the displacement, the formula simplifies to $W = |\mathbf{F}| |\vec{d}|$.

A deep-sea submersible starts its descent from an initial position $Q_1(10, 20, 5)$ metres and travels in a straight line to a target location $Q_2(70, 60, 25)$ metres. The coordinates are relative to a fixed origin on the surface. The submersible's propulsion system generates a constant force of $150\,000$ N in the direction of motion. Using the simplified formula from part (b), calculate the total energy expended by the propulsion system during this descent. Give your answer to three significant figures.

(a) A robotic arm applies a force $F_1 = \begin{pmatrix} 3 \\ -2 \\ 5 \end{pmatrix}$ N to an object. The object exerts a reaction force $F_2 = \begin{pmatrix} k \\ 1 \\ -2 \end{pmatrix}$ N on the arm, where $k \in \mathbb{R}$.

Given that the force $F_1$ is perpendicular to the reaction force $F_2$, find the value of $k$.

(b) The torque, $\tau$, generated by the robotic arm is given by the vector equation $\tau = c \cdot r \times F_1$, where $r = \begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix}$ m is a position vector and $c \in \mathbb{R}^+$ is a positive constant.

Given that the magnitude of the torque is $|\tau| = 3\sqrt{2}$ Nm, find the value of $c$.

A drone is being tracked in a 3D coordinate system. At time $t_1$, its position is at point A$(1, -2, 3)$. At a later time $t_2$, its position is at point B$(5, 4, -1)$.

Find the distance the drone travelled from A to B.

Another drone's flight path is modelled by the line $L$ with vector equation $\boldsymbol{r} = \begin{pmatrix} 0 \\ 1 \\ 5 \end{pmatrix} + t \begin{pmatrix} 2 \\ 3 \\ -2 \end{pmatrix}$. A control tower observes this drone at point C$(6, y, -1))$ which lies on line $L$.

Find the value of $y$.

Using the value of $y$ found in part (b), the position of point C is $(6, 10, -1))$.

Show that $\vec{AC} = \begin{pmatrix} 5 \\ 12 \\ -4 \end{pmatrix}$.

Find the angle between the drone's flight segment $\vec{AB}$ and the segment $\vec{AC}$. Give your answer in degrees to one decimal place.

Find the area of the triangle formed by the points A, B, and C. Give your answer to three significant figures.

The position vectors of two particles, P and Q, relative to an origin O are given by $p = \begin{pmatrix} 2 \\ k \\ 1 \end{pmatrix}$ and $q = \begin{pmatrix} k \\ -1 \\ 2k^2 \end{pmatrix}$ respectively, where $k \in \mathbb{R}$.

(a) Find an expression for the scalar product $p \bullet q$ in terms of $k$.

(b) Hence, find the set of values for $k$ for which the angle between the position vectors $p$ and $q$ is acute.

A mechanic applies a force, $\boldsymbol{F} = \begin{pmatrix} 10 \\ k \\ -50 \end{pmatrix}$ N, to the end of a wrench. The direction of this force is perpendicular to a vector $\boldsymbol{d} = \begin{pmatrix} 3 \\ 5 \\ 1 \end{pmatrix}$.

(a) Find the value of $k$.

The wrench applies the force at a point P, which has a position vector $\boldsymbol{r} = \begin{pmatrix} 0.2 \\ 0.1 \\ 0 \end{pmatrix}$ m relative to the bolt (which is at the origin). The torque, $\boldsymbol{\tau}$, is given by the vector equation $\boldsymbol{\tau} = \boldsymbol{r} \times \boldsymbol{F}$. The actual force applied is given by $\boldsymbol{F} = c \begin{pmatrix} 10 \\ 4 \\ -50 \end{pmatrix}$, where $c \in \mathbb{R}^+$.

(b) Given that the magnitude of the torque is 20 Nm, find the value of $c$.