Mathematical Methods for Physics and Engineering: A Comprehensive Guide

K. F. Riley, M. P. Hobson, S. J. Bence

Chapter 7 Vector algebra - all with Video Answers

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Chapter Questions

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Problem 1

Which of the following statements about general vectors $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ are true?
(a) $\mathbf{c} \cdot(\mathbf{a} \times \mathbf{b})=(\mathbf{b} \times \mathbf{a}) \cdot \mathbf{c}$.
(b) $\mathbf{a} \times(\mathbf{b} \times \mathbf{c})=(\mathbf{a} \times \mathbf{b}) \times \mathbf{c}$.
(c) $\mathbf{a} \times(\mathbf{b} \times \mathbf{c})=(\mathbf{a} \cdot \mathbf{c}) \mathbf{b}-(\mathrm{a} \cdot \mathbf{b}) \mathbf{c}$.
(d) $\mathrm{d}=\lambda \mathrm{a}+\mu \mathrm{b}$ implies $(\mathbf{a} \times \mathbf{b}) \cdot \mathbf{d}=0$
(e) $\mathbf{a} \times \mathbf{c}=\mathbf{b} \times \mathbf{c}$ implies $\mathbf{c} \cdot \mathbf{a}-\mathbf{c} \cdot \mathbf{b}=c|\mathbf{a}-\mathbf{b}|$
(f) $(a \times b) \times(c \times b)=b[b \cdot(c \times a)]$

Victor Salazar

Numerade Educator

01:53

Problem 2

A unit cell of diamond is a cube of side $A$ with carbon atoms at each corner, at the centre of each face and, in addition, displaced by $\frac{1}{4} A(\mathbf{i}+\mathbf{j}+\mathbf{k})$ from each of the previously mentioned ones, where $\mathbf{i}, \mathbf{j}, \mathbf{k}$ are unit vectors along the cube axes. One corner of the cube is taken as the origin of coordinates. What are the vectors joining the atom at $\frac{1}{4} A(\mathbf{i}+\mathbf{j}+\mathbf{k})$ to its four nearest neighbours? Determine the angle between the carbon bonds in diamond.

Penny Riley

Numerade Educator

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Problem 3

Identify the following surfaces:
(a) $|\mathbf{r}|=k ;$ (b) $\mathbf{r} \cdot \mathbf{u}=l ;$ (c) $\mathbf{r} \cdot \mathbf{u}=m|\mathbf{r}|$ for $-1 \leq m \leq+1$;
(d) $|\mathbf{r}-(\mathbf{r} \cdot \mathbf{u}) \mathbf{u}|=n$
Here $k$. I. $m$ and $n$ are fixed scalars and $u$ is a fixed unit yector.

Victor Salazar

Numerade Educator

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Problem 4

Find the angle between the position vectors to the points $(3,-4,0)$ and $(-2,1,0)$ and find the direction cosines of a vector perpendicular to both.

Victor Salazar

Numerade Educator

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Problem 5

$A, B, C$ and $D$ are the four corners, in order, of one face of a cube of side 2 units. The opposite face has corners $E, F, G$ and $H$, with $A E, B F, C G$ and $D H$ as parallel edges of the cube. The centre $O$ of the cube is taken as the origin and the $x-, y$ - and $z$-axes are parallel to $A D, A E$ and $A B$ respectively. Find the following:
(a) the angle between the face diagonal $A F$ and the body diagonal $A G$;
(b) the equation of the plane through $B$ that is parallel to the plane $C G E$;
(c) the perpendicular distance from the centre $J$ of the face $B C G F$ to the plane $O C G$
(d) the volume of the tetrahedron $J O C G$.

Victor Salazar

Numerade Educator

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Problem 6

Use vector methods to prove that the lines joining the mid-points of the opposite edges of a tetrahedron $O A B C$ meet at a point and that this point bisects each of the lines.

Eduard Sanchez

Numerade Educator

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Problem 8

Prove, by writing it out in component form, that
$$
(a \times b) \times c=(a \cdot c) \mathbf{b}-(b \cdot c) a
$$
and deduce the result, stated in (7.25), that the operation of forming the vector product is non-associative.

Eduard Sanchez

Numerade Educator

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Problem 9

Prove Lagrange's identity, i.e.
$$
(\mathbf{a} \times \mathbf{b}) \cdot(\mathbf{c} \times \mathbf{d})=(\mathbf{a} \cdot \mathbf{c})(\mathbf{b}-\mathbf{d})-(\mathbf{a} \cdot \mathbf{d})(\mathbf{b} \cdot \mathbf{c})
$$

Eduard Sanchez

Numerade Educator

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Problem 10

For four arbitrary vectors a, b, $\mathbf{c}$ and d, evaluate
$$
(\mathbf{a} \times \mathbf{b}) \times(\mathbf{c} \times \mathbf{d})
$$
in two different ways and so prove that
$$
\mathbf{a}[\mathbf{b}, \mathbf{c}, \mathbf{d}]-\mathbf{b}[\mathbf{c}, \mathbf{d}, \mathbf{a}]+\mathbf{c}[\mathbf{d}, \mathbf{a}, \mathbf{b}]-\mathbf{d}[\mathbf{a}, \mathbf{b}, \mathbf{c}]=0
$$
Show that this reduces to the normal Cartesian representation of the vector d, i.e. $d_{x} \mathbf{i}+d_{y} \mathbf{j}+d_{z} \mathbf{k}$ if $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ are taken as $\mathbf{i}, \mathbf{j}$ and $\mathbf{k}$, the Cartesian base vectors.

Eduard Sanchez

Numerade Educator

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Problem 11

Show that the points $(1,0,1),(1,1,0)$ and $(1,-3,4)$ lie on a straight line. Give the equation of the line in the form
$$
\mathbf{r}=\mathbf{a}+\lambda \mathbf{b}
$$

Rashmi Sinha

Numerade Educator

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Problem 12

The plane $P_{1}$ contains the points $A, B$ and $C$, which have position vectors $\mathbf{a}=-3 \mathbf{i}+2 \mathbf{j}, \mathbf{b}=7 \mathbf{i}+2 \mathbf{j}$ and $\mathbf{c}=2 \mathbf{i}+3 \mathbf{j}+2 \mathbf{k}$ respectively. Plane $P_{2}$ passes through $A$ and is orthogonal to the line $B C$, whilst plane $P_{3}$ passes through $B$ and is orthogonal to the line $A C$. Find the coordinates of $\mathbf{r}$, the point of intersection of the three planes.

Victor Salazar

Numerade Educator

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Problem 13

Two planes have non-parallel unit normals $\hat{\mathbf{n}}$ and $\hat{m}$ and their closest distances from the origin are $\lambda$ and $\mu$ respectively. Find the vector equation of their line of intersection in the form $\mathbf{r}=v \mathbf{p}+\mathbf{a}$.

Victor Salazar

Numerade Educator

04:19

Problem 14

Two fixed points, $A$ and $B$, in three-dimensional space have position vectors a and b. Identify the plane $P$ given by
$$
(\mathbf{a}-\mathbf{b})-\mathbf{r}=\frac{1}{2}\left(a^{2}-b^{2}\right)
$$
where $a$ and $b$ are the magnitudes of a and $b$.
Show also that the equation
$$
(\mathbf{a}-\mathbf{r}) \cdot(\mathbf{b}-\mathbf{r})=0
$$
describes a sphere $S$ of radius $|\mathbf{a}-\mathbf{b}| / 2 .$ Deduce that the intersection of $P$ and $S$ is also the intersection of two spheres, centred on $A$ and $B$ and each of radius $|a-b| f^{2}$

Wen Zheng

Numerade Educator

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Problem 15

Let $O, A, B$ and $C$ be four points with position vectors $0, a, b$ and $c$, and denote by $\mathrm{g}=\lambda \mathrm{a}+\mu \mathrm{b}+\mathrm{vc}$ the position of the centre of the sphere on which they all lie.
(a) Prove that $\lambda, \mu$ and $v$ simultaneously satisfy
$$
(\mathbf{a} \cdot \mathbf{a}) \hat{\lambda}+(\mathbf{a}-\mathbf{b}) \mu+(\mathbf{a}-\mathbf{c}) v=\frac{1}{2} a^{2}
$$
and two other similar equations.
(b) By making a change of origin, find the centre and radius of the sphere on which the points $\mathbf{p}=3 \mathbf{i}+\mathbf{j}-2 \mathbf{k}, \mathbf{q}=4 \mathbf{i}+3 \mathbf{j}-3 \mathbf{k}, \mathbf{r}=7 \mathbf{i}-3 \mathbf{k}$ and $\mathbf{s}=6 \mathbf{i}+\mathbf{j}-\mathbf{k}$ all lic.

Eduard Sanchez

Numerade Educator

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Problem 16

The vectors $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ are coplanar and related by
$$
\lambda \mathbf{a}+\mu \mathbf{b}+v \mathbf{c}=0
$$
where $\lambda, \mu, v$ are not all zero. Show that the condition for the points with position vectors $\alpha \mathbf{a}, \beta \mathbf{b}$ and $\gamma \mathrm{c}$ to be collinear is
$$
\frac{\lambda}{\alpha}+\frac{\mu}{\beta}+\frac{v}{\gamma}=0
$$

Eduard Sanchez

Numerade Educator

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Problem 17

(a) Show that the line of intersection of the planes $x+2 y+3 z=0$ and $3 x+2 y+z=0$ is cqually inclined to the $x$ - and $z-$ axes and makes an angle $\cos ^{-1}(-2 / \sqrt{6})$ with the $y$-axis.
(b) Find the perpendicular distance between one corner of a unit cube and the major diagonal not passing through it.

Eduard Sanchez

Numerade Educator

03:45

Problem 18

Four points $X_{i}(i=1,2,3,4)$, taken for simplicity as all lying within the octant $x, y, z \geq 0$, have position vectors $\mathbf{x}_{i}$. Convince yourself that vector $\mathbf{x}_{\mathrm{n}}$ lies within the sector of space defined by the other three vectors if
$$
\max _{\text {aer } i}\left\{\min _{\operatorname{aver} j+i}\left[\frac{\mathbf{x}_{i} \cdot \mathbf{x}_{i}}{\left|\mathbf{x}_{i}\right|\left|\mathbf{x}_{j}\right|}\right]\right\}=n
$$
i.e. if $n$ cquals that value of $i$ for which the largest of the set of angles which $\mathbf{x}_{\mathrm{i}}$ makes with the other vectors is the lowest. Determine whether any of the four points with coordinates
$$
X_{1}=(3,2,2), \quad X_{2}=(2,3,1), \quad X_{3}=(2,1,3), \quad X_{4}=(3,0,3)
$$

Darshan Maheshwari

Numerade Educator

02:00

Problem 19

The vectors $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ are not coplanar. The vectors $\mathbf{a}^{\prime}, \mathbf{b}$ ' and $\mathbf{c}^{\prime}$ are theassociated reciprocal vectors. Verify that the expressions $(7.49)-(7.51)$ define a set of reciprocal vectors $a^{\prime}, \mathbf{b}^{\prime}$ and $\mathbf{c}^{\prime}$ with the following properties:
(a) $\mathbf{a}^{\prime}-\mathbf{a}=\mathbf{b}^{\prime} \cdot \mathbf{b}=\mathbf{c}^{\prime} \cdot \mathbf{c}=1$
(b) $\mathbf{a}^{\prime}-\mathbf{b}=\mathbf{a}^{\prime} \cdot \mathbf{c}=\mathbf{b}^{\prime} \cdot \mathbf{a} \quad$ etc $=0 ;$
(c) $\left[\mathbf{a}^{\prime}, \mathbf{b}^{\prime}, \mathbf{c}^{\prime}\right]=1 /[\mathbf{a}, \mathbf{b}, \mathbf{c}] ;$
(d) $\mathbf{a}=\left(\mathbf{b}^{\prime} \times \mathbf{c}^{\prime}\right) /\left[\mathbf{a}^{\prime}, \mathbf{b}^{\prime}, \mathbf{c}^{\prime}\right]$.

Ankit Singh

Numerade Educator

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Problem 20

Three non-coplanar vectors $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$, have as their respective reciprocal vectors the set $\mathbf{a}^{\prime}, \mathbf{b}$ ' and $\mathbf{c}^{\prime}$. Show that the normal to the plane containing the points $k^{-1} \mathbf{a}, l^{-1} \mathbf{b}$ and $m^{-1} \mathbf{c}$ is in the direction of the vector $k a^{\prime}+l b^{\prime}+m c^{\prime} .$

Eduard Sanchez

Numerade Educator

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Problem 21

Victor Salazar

Numerade Educator

05:37

Problem 22

In a crystal with a face-centred cubic structure, the basic cell can be taken as a cube of edge $a$ with its centre at the origin of coordinates and its edges parallel to the Cartesian coordinate axes; atoms are sited at the eight corners and at the centre of each face. However, other basic cells are possible. One is the rhomboid shown in figure $7.17$, which has the three vectors $b, c$ and $d$ as edges.
(a) Show that the volume of the rhomboid is one-quarter that of the cube.
(b) Show that the angles between pairs of edges of the rhomboid are $60^{\circ}$ and that the corresponding angles between pairs of edges of the rhomboid defined by the reciprocal vectors to b, c, d are cach $109.5^{\circ}$. (This rhomboid can be used as the basic cell of a body-centred cubic structure, more casily visualised as a cube with an atom at each corner and one at its centre.)
(c) In order to use the Bragg formula, $2 d \sin \theta=n \hat{\lambda}$, for the scattering of X-rays by a crystal, it is necessary to know the perpendicular distance $d$ between successive planes of atoms; for a given crystal structure, $d$ has a particular value for each set of planes considered. For the face-centred cubic structure find the distance between suocessive planes with normals in the $\mathbf{k}, \mathbf{i}+\mathbf{j}$ and $\mathbf{i}+\mathbf{j}+\mathbf{k}$ directions.from the origin is given by $m|\mathbf{v}| d$. Show that if $\mathbf{r}$ is the position of the particle then the vector $\mathbf{J}=\mathbf{r} \times$ mv represents the angular momentum.
(b) Now consider a rigid collection of particles (or a solid body) rotating about an axis through the origin, the angular velocity of the collection being represented by $\omega$.
(i) Show that the velocity of the ith particle is
$$
\mathbf{v}_{i}=\boldsymbol{c} \times \mathbf{r}_{i}
$$
and that the total angular momentum $\mathbf{J}$ is
$$
\mathbf{J}=\sum_{i} m_{i}\left[r_{i}^{2} \omega-\left(\mathbf{r}_{i}-\omega\right) \mathbf{r}_{i}\right]
$$
(ii) Show further that the component of J along the axis of rotation can be written as $I \omega$, where $I$, the moment of inertia of the collection about the axis or rotation, is given by
$$
I=\sum_{i} m_{i} \rho_{i}^{2}
$$
Interpret $\rho_{\mathrm{i}}$ geometrically.
(iii) Prove that the total kinetic energy of the particles is $\frac{1}{2} I \omega^{2}$.

Mayukh Banik

Numerade Educator

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Problem 23

By proceeding as indicated below, prove the parallel axis theorem, which states that, for a body of mass $M$, the moment of inertia $I$ about any axis is related to the corresponding moment of inertia $I_{0}$ about a parallel axis that passes through the centre of mass of the body by
$$
I=I_{0}+M a_{\perp}^{2}
$$
where $a_{\perp}$ is the perpendicular distance between the two axes. Note that $I_{0}$ can be written as
$$
\int(\hat{\mathbf{n}} \times \mathbf{r}) \cdot(\mathbf{\mathbf { n }} \times \mathbf{r}) d m
$$
where $\mathbf{r}$ is the vector position, relative to the centre of mass, of the infinitesimal mass $d m$ and $\hat{n}$ is a unit vector in the direction of the axis of rotation. Write a similar expression for $I$ in which $\mathbf{r}$ is replaced by $\mathbf{r}^{\prime}=\mathbf{r}-\mathbf{a}$, where $\mathbf{a}$ is the vector position of any point on the axis to which $I$ refers. Use Lagrange's identity and the fact that $\int \mathbf{r} d m=0$ (by the definition of the centre of mass) to establish the result

Victor Salazar

Numerade Educator

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Problem 24

Without carrying out any further integration, use the results of the previous exercise, the worked example in subsection $6.3 .4$ and exercise $6.10$ to prove that the moment of inertia of a uniform rectangular lamina, of mass $M$ and sides $a$ and $b$, about an axis perpendicular to its plane and passing through the point $(\alpha a / 2, \beta b / 2)$, with $-1 \leq x, \beta \leq 1$, is
$$
\frac{M}{12}\left[a^{2}\left(1+3 x^{2}\right)+b^{2}\left(1+3 \beta^{2}\right)\right]
$$

Eduard Sanchez

Numerade Educator

04:44

Problem 26

Systems that can be modelled as damped harmonic oscillators are widespread; pendulum clocks, car shock absorbers, tuning circuits in television sets and radios, and collective electron motions in plasmas and metals are just a few examples.
In all these cases, one or more variables describing the system obey(s) an equation of the form
$$
\ddot{x}+2 \gamma \dot{x}+\omega_{0}^{2} x=P \cos \omega t
$$
where $x=d x / d t$, etc. and the inclusion of the factor 2 is conventional. In the steady state (i.e. after the effects of any initial displacement or velocity have been damped out) the solution of the equation takes the form
$$
x(t)=A \cos (\omega t+\phi)
$$
By expressing each term in the form $B \cos (\omega t+\epsilon)$ and representing it by a vector of magnitude $B$ making an angle $\epsilon$ with the $x$-axis, draw a closed vector diagram, at $t=0$, say, that is equivalent to the equation.
(a) Convince yourself that whatever the value of $\omega(>0) \phi$ must be negative $(-\pi<\phi \leq 0)$ and that
$$
\phi=\tan ^{-1}\left(\frac{-2 \gamma \omega}{\omega_{0}^{2}-\omega^{2}}\right)
$$
(b) Obtain an expression for $A$ in terms of $P, \omega_{0}$ and $\omega$.

Ajay Singhal

Numerade Educator

07:34

Problem 27

According to alternating current theory, the currents and voltages in the components of the circuit shown in figure $7.18$ are determined by Kirchhoff's laws and the relationships
$$
I_{1}=\frac{V_{1}}{R_{1}}, \quad I_{2}=\frac{V_{2}}{R_{2}}, \quad I_{3}=i \omega C V_{3}, \quad V_{4}=i \omega L I_{2}
$$
The factor $i=\sqrt{-1}$ in the expression for $I_{3}$ indicates that the phase of $I_{3}$ is $90^{\circ}$ ahead of $V_{3}$. Similarly the phase of $V_{4}$ is $90^{\circ}$ ahead of $I_{2}$.

Measurement shows that $V_{3}$ has an amplitude of $0.661 V_{0}$ and a phase of $+13.4^{\circ}$ relative to that of the power supply. Taking $V_{0}=1 \mathrm{~V}$ and using a series of vector plots for voltages and currents (they could all be on the same plot if suitable scales were chosen), determine all unknown currents and voltages and find values for the inductance of $L$ and the resistance of $R_{2}$. (Scales of $1 \mathrm{~cm}=$ $0.1 \mathrm{~V}$ for voltages and $1 \mathrm{~cm}=1 \mathrm{~mA}$ for currents are convenient.)

Rishi Rao

Numerade Educator