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

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.

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 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.

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 temporary exhibition structure is being designed. The coordinates of three points on the ground are P(0,0,0), Q(8,0,0), and R(0,6,0). A supporting mast is erected at point S(4,3,10).

(a) Calculate the length of the structural cable $SQ$.

(b) Determine the angle Q$\hat{S}$R, in radians.

(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 $\Pi_3$.

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 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 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).

(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 $\mathbf{F_A}$.

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'.

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.

The vectors $\mathbf{u}$ and $\mathbf{v}$ are given by $\mathbf{u} = \begin{pmatrix} k \\ kr \\ kr^2 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} kr^2 \\ kr \\ k \end{pmatrix}$, where $k, r \in \mathbb{R}^+$.

Show that $\cos\theta = \frac{3r^2}{1+r^2+r^4}$, where $\theta$ is the angle between $\mathbf{u}$ and $\mathbf{v}$.

Given that the angle between $\mathbf{u}$ and $\mathbf{v}$ is $60^\circ$, find the possible values of $r^2$.

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.