Exercises

VECTOR ALGEBRA
not coplanar. Moreover, if a, b and c are mutually orthogonal unit vectors then
a = a, b = b and c = c, so that the two systems of vectors are identical.
Construct the reciprocal vectors of a = 2i, b = j + k, c = i + k.
First we evaluate the triple scalar product:
a · (b × c) = 2i · [(j + k) × (i + k)]
= 2i · (i + j − k) = 2.
Now we find the reciprocal vectors:
a = 12 (j + k) × (i + k) =
b =
c =
1
(i + k) × 2i = j,
2
1
(2i) × (j + k) = −j
2
1
(i
2
+ j − k),
+ k.
It is easily verified that these reciprocal vectors satisfy their defining properties (7.47),
(7.48). We may also use the concept of reciprocal vectors to define the components of a
vector a with respect to basis vectors e1 , e2 , e3 that are not mutually orthogonal.
If the basis vectors are of unit length and mutually orthogonal, such as the
Cartesian basis vectors i, j, k, then (see the text preceeding (7.21)) the vector a
can be written in the form
a = (a · i)i + (a · j)j + (a · k)k.
If the basis is not orthonormal, however, then this is no longer true. Nevertheless,
we may write the components of a with respect to a non-orthonormal basis
e1 , e2 , e3 in terms of its reciprocal basis vectors e1 , e2 , e3 , which are defined as in
(7.49)–(7.51). If we let
a = a1 e1 + a2 e2 + a3 e3 ,
then the scalar product a ·
e1
is given by
a · e1 = a1 e1 · e1 + a2 e2 · e1 + a3 e3 · e1 = a1 ,
where we have used the relations (7.48). Similarly, a2 = a·e2 and a3 = a·e3 ; so now
a = (a · e1 )e1 + (a · e2 )e2 + (a · e3 )e3 .
(7.52)
7.10 Exercises
7.1
Which of the following statements about general vectors a, b and c are true?
(a)
(b)
(c)
(d)
(e)
(f)
c · (a × b) = (b × a) · c.
a × (b × c) = (a × b) × c.
a × (b × c) = (a · c)b − (a · b)c.
d = λa + µb implies (a × b) · d = 0.
a × c = b × c implies c · a − c · b = c|a − b|.
(a × b) × (c × b) = b[b · (c × a)].
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7.10 EXERCISES
7.2
7.3
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, at positions displaced by 14 A(i + j + k)
from each of those already mentioned; i, j, 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 14 A(i + j + k) to its four nearest neighbours? Determine the
angle between the carbon bonds in diamond.
Identify the following surfaces:
(a) |r| = k; (b) r · u = l; (c) r · u = m|r| for −1 ≤ m ≤ +1;
(d) |r − (r · u)u| = n.
7.4
7.5
Here k, l, m and n are fixed scalars and u is a fixed unit vector.
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.
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 AE, BF, CG and DH
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 AD, AE and AB, respectively. Find the
following:
(a) the angle between the face diagonal AF and the body diagonal AG;
(b) the equation of the plane through B that is parallel to the plane CGE;
(c) the perpendicular distance from the centre J of the face BCGF to the plane
OCG;
(d) the volume of the tetrahedron JOCG.
7.6
7.7
7.8
Use vector methods to prove that the lines joining the mid-points of the opposite
edges of a tetrahedron OABC meet at a point and that this point bisects each of
the lines.
The edges OP , OQ and OR of a tetrahedron OP QR are vectors p, q and r,
respectively, where p = 2i + 4j, q = 2i − j + 3k and r = 4i − 2j + 5k. Show that
OP is perpendicular to the plane containing OQR. Express the volume of the
tetrahedron in terms of p, q and r and hence calculate the volume.
Prove, by writing it out in component form, that
(a × b) × c = (a · c)b − (b · c)a,
7.9
and deduce the result, stated in equation (7.25), that the operation of forming
the vector product is non-associative.
Prove Lagrange’s identity, i.e.
(a × b) · (c × d) = (a · c)(b · d) − (a · d)(b · c).
7.10
For four arbitrary vectors a, b, c and d, evaluate
(a × b) × (c × d)
in two different ways and so prove that
a[b, c, d] − b[c, d, a] + c[d, a, b] − d[a, b, c] = 0.
7.11
Show that this reduces to the normal Cartesian representation of the vector d,
i.e. dx i + dy j + dz k, if a, b and c are taken as i, j and k, the Cartesian base vectors.
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
r = a + λb.
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VECTOR ALGEBRA
7.12
7.13
7.14
The plane P1 contains the points A, B and C, which have position vectors
a = −3i + 2j, b = 7i + 2j and c = 2i + 3j + 2k, respectively. Plane P2 passes
through A and is orthogonal to the line BC, whilst plane P3 passes through
B and is orthogonal to the line AC. Find the coordinates of r, the point of
intersection of the three planes.
Two planes have non-parallel unit normals n̂ and m̂ and their closest distances
from the origin are λ and µ, respectively. Find the vector equation of their line
of intersection in the form r = νp + a.
Two fixed points, A and B, in three-dimensional space have position vectors a
and b. Identify the plane P given by
(a − b) · r = 12 (a2 − b2 ),
where a and b are the magnitudes of a and b.
Show also that the equation
(a − r) · (b − r) = 0
7.15
describes a sphere S of radius |a − 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|/ 2.
Let O, A, B and C be four points with position vectors 0, a, b and c, and denote
by g = λa + µb + νc the position of the centre of the sphere on which they all lie.
(a) Prove that λ, µ and ν simultaneously satisfy
(a · a)λ + (a · b)µ + (a · c)ν = 12 a2
and two other similar equations.
(b) By making a change of origin, find the centre and radius of the sphere on
which the points p = 3i + j − 2k, q = 4i + 3j − 3k, r = 7i − 3k and s = 6i + j − k
all lie.
7.16
The vectors a, b and c are coplanar and related by
λa + µb + νc = 0,
where λ, µ, ν are not all zero. Show that the condition for the points with position
vectors αa, βb and γc to be collinear is
λ
µ
ν
+ + = 0.
α β
γ
7.17
Using vector methods:
(a) Show that the line of intersection of the planes x + 2y + 3z = 0 and
3x + 2y + √
z = 0 is equally inclined to the x- and z-axes and makes an angle
cos−1 (−2/ 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.
7.18
Four points Xi , i = 1, 2, 3, 4, taken for simplicity as all lying within the octant
x, y, z ≥ 0, have position vectors xi . Convince yourself that the direction of
vector xn lies within the sector of space defined by the directions of the other
three vectors if
xi · xj
,
min
over j |xi ||xj |
considered for i = 1, 2, 3, 4 in turn, takes its maximum value for i = n, i.e. n equals
that value of i for which the largest of the set of angles which xi makes with
the other vectors, is found to be the lowest. Determine whether any of the four
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7.10 EXERCISES
a
b
c
d
a
Figure 7.17 A face-centred cubic crystal.
points with coordinates
X1 = (3, 2, 2),
7.19
7.21
X3 = (2, 1, 3),
X4 = (3, 0, 3)
lies within the tetrahedron defined by the origin and the other three points.
The vectors a, b and c are not coplanar. The vectors a , b and c are the
associated reciprocal vectors. Verify that the expressions (7.49)–(7.51) define a set
of reciprocal vectors a , b and c with the following properties:
(a)
(b)
(c)
(d)
7.20
X2 = (2, 3, 1),
a · a = b · b = c · c = 1;
a · b = a · c = b · a etc = 0;
[a , b , c ] = 1/[a, b, c];
a = (b × c )/[a , b , c ].
Three non-coplanar vectors a, b and c, have as their respective reciprocal vectors
the set a , b and c . Show that the normal to the plane containing the points
k −1 a, l −1 b and m−1 c is in the direction of the vector ka + lb + mc .
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◦ and that
the corresponding angles between pairs of edges of the rhomboid defined by
the reciprocal vectors to b, c, d are each 109.5◦ . (This rhomboid can be used
as the basic cell of a body-centred cubic structure, more easily visualised as
a cube with an atom at each corner and one at its centre.)
(c) In order to use the Bragg formula, 2d sin θ = nλ, 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 successive planes with normals in the k, i + j and
i + j + k directions.
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VECTOR ALGEBRA
7.22
In subsection 7.6.2 we showed how the moment or torque of a force about an axis
could be represented by a vector in the direction of the axis. The magnitude of
the vector gives the size of the moment and the sign of the vector gives the sense.
Similar representations can be used for angular velocities and angular momenta.
(a) The magnitude of the angular momentum about the origin of a particle of
mass m moving with velocity v on a path that is a perpendicular distance d
from the origin is given by m|v|d. Show that if r is the position of the particle
then the vector J = r × 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 ω.
(i) Show that the velocity of the ith particle is
vi = ω × ri
and that the total angular momentum J is
mi [ri2 ω − (ri · ω)ri ].
J=
i
(ii) Show further that the component of J along the axis of rotation can
be written as Iω, where I, the moment of inertia of the collection
about the axis or rotation, is given by
I=
mi ρ2i .
i
Interpret ρi geometrically.
(iii) Prove that the total kinetic energy of the particles is 12 Iω 2 .
7.23
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 I0 about a parallel axis that passes through
the centre of mass of the body by
I = I0 + Ma2⊥ ,
where a⊥ is the perpendicular distance between the two axes. Note that I0 can
be written as
(n̂ × r) · (n̂ × r) dm,
7.24
where r is the vector position, relative to the centre of mass, of the infinitesimal
mass dm and n̂ is a unit vector in the direction of the axis of rotation. Write a
similar expression for I in which r is replaced by r = r − a, where a is the vector
position of any
point on the axis to which I refers. Use Lagrange’s identity and
the fact that r dm = 0 (by the definition of the centre of mass) to establish the
result.
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
(αa/2, βb/2), with −1 ≤ α, β ≤ 1, is
M 2
[a (1 + 3α2 ) + b2 (1 + 3β 2 )].
12
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7.10 EXERCISES
V1
R1 = 50 Ω I2
I1
I3
V4
V2
L
R2
C = 10 µF
V0 cos ωt
V3
Figure 7.18 An oscillatory electric circuit. The power supply has angular
frequency ω = 2πf = 400π s−1 .
7.25
Define a set of (non-orthogonal) base vectors a = j + k, b = i + k and c = i + j.
(a) Establish their reciprocal vectors and hence express the vectors p = 3i−2j+k,
q = i + 4j and r = −2i + j + k in terms of the base vectors a, b and c.
(b) Verify that the scalar product p · q has the same value, −5, when evaluated
using either set of components.
7.26
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
ẍ + 2γẋ + ω02 x = P cos ωt,
where ẋ = dx/dt, 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(ωt + φ).
By expressing each term in the form B cos(ω t + ), and representing it by a vector
of magnitude B making an angle 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 ω (> 0) φ must be negative
(−π < φ ≤ 0) and that
−2γω
φ = tan−1
.
2
ω0 − ω 2
(b) Obtain an expression for A in terms of P , ω0 and ω.
7.27
According to alternating current theory, the currents and potential differences in
the components of the circuit shown in figure 7.18 are determined by Kirchhoff’s
laws and the relationships
I1 =
√
V1
,
R1
I2 =
V2
,
R2
I3 = iωCV3 ,
V4 = iωLI2 .
The factor i = −1 in the expression for I3 indicates that the phase of I3 is 90◦
ahead of V3 . Similarly the phase of V4 is 90◦ ahead of I2 .
Measurement shows that V3 has an amplitude of 0.661V0 and a phase of
+13.4◦ relative to that of the power supply. Taking V0 = 1 V, and using a series
239