8.1.1 Vector algebra

Scalars are manipulated by the rules of ordinary algebra, which we now call scalar algebra, that were discussed in Section 1.2.1. Vectors are manipulated by an analogous set of rules known as vector algebra, which we will explore in this chapter. We note that the rules differ from those of scalar algebra in several respects: for example, the commutative law of multiplication for scalars does not necessarily hold for vectors; also, if the product of two vectors is zero, this does not imply that one, or both, of them is necessarily zero.

We begin by looking at addition and subtraction of vectors. In Figure 8.2 we added vector b to vector a to obtain the vector s = a + b. As can be seen from Figure 8.3a, if we instead add a to b, we obtain the same result. In other words, addition obeys the commutative law

(8.1a)

The triangle law is also referred to as the parallelogram law, because the construction of Figure 8.3a produces a parallelogram. Likewise, Figure 8.3b shows the construction for the difference d = a − b, corresponding to adding the vector −b to a. As in the case of scalar algebra, subtraction is not commutative: a − b ≠ b − a. Also note that |s| ≠ |a| + |b| and |d| ≠ |a| − |b|. The magnitudes of s and d are found from Figure 8.3 by using the cosine rule derived in Chapter 2. These constructions may be extended to more than two vectors and one easily establishes the associative law

(8.1b)

using the constructions of Figure 8.4.

Products of vectors will be discussed in Section 8.2. To complete this subsection, we consider the product λa of a scalar λ with a vector a. If λ > 0, this is defined to be a vector of magnitude λ|a| in the direction of a; and if λ < 0, it is defined to be a vector of length |λ||a| in the opposite direction to a, as illustrated in Figure 8.5. Division by a scalar λ is defined as multiplication by λ^{− 1} and in both cases the operations are associative and distributive, so that

(8.2a)

(8.2b)

and

(8.2c)

Finally, it is often useful to introduce the null vector 0, which has zero length, so that, for example,

for any vector a. Vectors with unit magnitude, called unit vectors, also play a special role and are obtained by dividing a vector by its magnitude, a procedure called normalisation. Unit vectors in the same directions as a, b, … are usually denoted ; thus .

8.1.2 Components of vectors: Cartesian co-ordinates

A useful representation of vectors is obtained by using Cartesian co-ordinates, defined with respect to a right-handed set of axes, as shown in Figure 1.9 and again in Figure 8.7a. Then, using the triangle law of addition, an ordinary vector a can always be written as the sum

of three vectors a_x, a_y, a_z parallel to the x-, y- and z-axes, respectively, as shown in Figure 8.7b. If we now introduce three unit vectors i, j, k in the x, y and z directions, respectively, then

and

(8.3a)

where

(8.3b)

as can be seen by applying Pythagoras' theorem twice to Figure 8.7b.

The vectors i, j and k are called the basis vectors of the Cartesian system¹ and a_x, a_y and a_z are called the components of the vector a in the x, y and z directions.² They are useful because they enable a vector a to be defined completely by its components without the necessity of drawing a diagram to specify its direction. They also enable equations involving vectors to be expressed as three equations for their components. This is analogous to the situation we met in Chapter 6 for complex variables, where an equation involving complex quantities could be written as two equations involving real variables. For example, the sum s of two vectors a and b, where

is, using (8.1) and (8.2),

so the three equations for the components are

In Section 1.3.2, a point in a plane was specified by its x and y co-ordinates, defined as its projections onto the x- and y-axes. The description extends directly to three dimensions when a point in space P(x, y, z) is specified by three co-ordinates in the same way. However, in many applications it is useful to describe it using a position vector r, which translates the origin to a point P, as shown in Figure 8.8. This can be decomposed into its components in exactly the same way as the vector a of Figure 8.7, when one obtains

(8.4a)

In other words, the components of the position vector are just the Cartesian co-ordinates (x, y, z), and its magnitude, or length,

(8.4b)

is just the distance of the point P from the origin of the co-ordinate system.

Another way of representing P is in terms of the angles its position vector makes with the x, y and z-axes. In Figure 8.9a, r is the position vector of the point P(x, y, z), with components x, y and z along the three axes and Figure 8.9b shows the three angles α, β, and γ that r makes with the x, y and z-axes, respectively. The ratios x: y: z are called the direction ratios of r and

(8.5a)

are its direction cosines. It follows from (8.4b) and (8.5a) that

(8.5b)

and the unit vector is given by

(8.5c)

in Cartesian co-ordinates.

So far, we have considered basis vectors i, j, k that are unit vectors at right angles to each other. However, cases sometimes occur, in crystallography for example, where it is advantageous to use basis vectors a, b, c, that do not satisfy these criteria. This is possible because, for any three non-zero vectors a, b, c, which are not parallel to the same plane, one can always write an arbitrary vector r in the form

(8.6a)

where λ, μ, ν are uniquely determined real numbers.

To prove this result, we use the construction of Figure 8.10. The line OO′ has a position vector r. The lines OA, OB and OC are parallel to a, b and c, respectively. Taken in pairs, these three lines then define planes parallel respectively to the planes defined by the pairs of vectors (a, b), (b, c) and (c, a). By the law of addition,

where λ, μ and ν are real numbers. To show they are unique, we assume that r can be written

(8.6b)

Then subtracting (8.6a) from (8.6b) gives

But the left-hand side is a vector parallel to a, whereas the right-hand side is a vector parallel to the plane (b, c). Since a, b and c are all non-zero and not all parallel to the same plane, this result can only be true if

Thus λ = λ′ and similarly, μ = μ′ and ν = ν′.

8.2.1 Scalar product

Consider a particle that undergoes a linear displacement d under the action of a force F at an angle θ to the direction of the displacement. The component of F in the direction of d is Fcos θ and the product Fdcos θ is the work done by the force. The work done is a scalar and Fdcos θ is an example of a scalar product. More generally, the scalar product of two vectors a and b is defined as

(8.7)

where θ is the angle between the directions of a and b. Because cosine is an even function, it is irrelevant in which direction the angle θ is measured. Because of the notation on the left-hand side of (8.7), the scalar product is also called the dot product of a and b. From the definition (8.7), we see that the commutative law

(8.8a)

holds for scalar products, and if λ, μ are arbitrary scalars, then

(8.8b)

Further, bcos θ is the projection of b onto the axis defined by a and acos θ is the projection of a onto the axis defined by b, so that (8.7) can be rewritten in the forms

Taken together with the triangle law of addition shown in Figure 8.2, this implies the associative law

(8.8c)

The algebraic laws (8.8) are similar to those of scalar algebra and

just as for scalar multiplication. However, the inverse statement is not necessarily true, since a · b = 0 if cos θ = 0, that is, if the two vectors are at right angles. So a · b = 0 does not imply that either a or b is necessarily zero, i.e.

This is a fundamental difference between scalar and vector algebra. If

(8.9)

the vectors a and b are said to be orthogonal. Also, if a = b, then

(8.10)

is the squared magnitude, or squared ‘length’, of the vector.

Applying the definition (8.7) to the three unit vectors i, j and k in the Cartesian system gives

(8.11a)

and

(8.11b)

The unit vectors i, j and k are both orthogonal and normalised and are referred to as an orthonormal set of basis vectors. If we now write the vectors a and b in terms of Cartesian co-ordinates, i.e.

then using (8.8) and (8.11) gives

(8.12)

8.2.2 Vector product

The vector product, also called the cross product, of two vectors a and b is written a∧b or a × b, (we will use the latter notation) and defined as

(8.13)

where θ is the angle measured from the direction of a to that of b and is a unit vector perpendicular to the plane containing the two vectors in a direction determined by the ‘right-hand screw rule’ as shown in Figure 8.11. Because sin ( − θ) = −sin θ, it follows that changing the order of the factors in the product changes its sign, that is, the cross product is anti-commutative:

(8.14a)

Note that

and

In addition, the definition (8.13) leads to

(8.14b)

and

(8.14c)

by analogy with (8.8b) (8.8c) for scalar products.

Applying the definition (8.13) to unit Cartesian vectors gives the useful results:

(8.15a)

and

(8.15b)

Note the order of the vectors i, j, k in (8.15a). If the order is different from this, a minus sign is required because of the anti-commutative property (8.14a), i.e.

We can now evaluate the vector product of any two vectors in terms of their Cartesian components. Using (8.14) and (8.15) gives

(8.16a)

The structure of this result is best brought out by relabelling i, j, k as so that the direction with which they are associated is explicit. The vector product (8.16a) then becomes

(8.16b)

The sign of each term in (8.16b) is easily memorised by introducing the idea of a cyclic permutation, which is also useful in other contexts. A cyclic permutation of any three objects a, b, c in order abc is obtained by removing an object from the end of the sequence and placing it at the beginning, any number of times. Thus abc, cab and bca are cyclic permutations of abc, while acb, bac and cba are non-cyclic permutations. Using this, one sees that (8.16b) is the sum of six terms, each of which is itself the product of three terms, where:

1. the suffices are all different; and (ii) the sign of the product is +1, or −1, depending on whether the order of the suffices is a cyclic permutation of x, y, z (i.e. xyz, zxy or yzx) or a non-cyclic permutation (i.e. xzy, yxz or zyx).

A physical example of a vector product is provided by considering a rigid body rotating with angular velocity , where the direction of corresponds to the axis of rotation. Consider a point P on the body with position vector r and angle θ between r and , as shown in Figure 8.12. The vector product is a vector of magnitude ωrsin θ and by the right-hand rule is in the plane perpendicular to the axis of rotation. Since rsin θ is the radius of the circle of rotation of P, this has the same magnitude and the same direction as the linear velocity v of P, that is, .

A second physical example is the torque, or moment, about a point O generated by a force F acting on an object at a point P, corresponding to a position vector r relative to O, as shown in Figure 8.13. This is given by

(8.17)

where is a unit vector perpendicular to r and F that specifies the direction of . The magnitude of the torque is often written in the form τ = Fd where d = rsin θ is the perpendicular distance from O to a straight line through P in the direction of the force, as shown in Figure 8.13. This line is called the line of action of F. Now suppose instead that the same force acts at a different point P′. Then the torque is unchanged, provided that P and P′ lie on the same line of action, as shown in Figure 8.13 which represents the plane defined by the point O and the line of action of F. This is because

by simple geometry, since remains a unit vector out of the plane and the magnitude of the torque is given by τ = Fr′sin θ′ = Frsin θ = Fd in both cases.

8.2.3 Triple products

The product of a vector c with a vector product a × b can produce either a scalar c · (a × b) or a vector c × (a × b). The former is called a triple scalar product and the latter a triple vector product. We will discuss each in turn.

(i) Triple scalar product

If we start by expressing a vector product in terms of the components of its vectors using (8.16a) and then form the scalar product with c using (8.12), one obtains

(8.18)

As in (8.16b), the sign of each term is +1(or −1) depending on whether the order of the suffices is a cyclic (or non-cyclic) permutation of x, y, z. This in turn implies that if we make a cyclic rearrangement of a, b and c, for example c · (a × b) → a · (b × c), the triple scalar product is unchanged; whereas if we make a non-cyclic rearrangement, for example c · (a × b) → a · (c × b), the sign of the triple scalar product is reversed. Introducing the shorthand notation

(8.19)

leads to the results

(8.20)

Furthermore, since a · (b × c) = (b × c) · a by (8.8a), we can use (8.19) to write

(8.21)

Thus, provided the order is maintained, the dot and cross are interchangeable.

The triple scalar product has a simple geometrical interpretation, as shown in Figure 8.15. The vector (b × c) is perpendicular to the plane defined by b and c and with magnitude BCsin θ, equal to the shaded area. The scalar product with a is the product of this area with the projection of a along (b × c). Thus |a · (b × c)| is the volume of the parallelopiped with edges a, b and c. When the three vectors lie in a plane, the volume of the parallelepiped is zero. Hence the condition for three vectors to be coplanar is that their triple scalar product vanishes.

(8.22a)

It also vanishes if any two of the vectors are identical.

(8.22b)

A physical example of a triple scalar product is the torque, or moment, about an axis (rather than about a point). Suppose F is a force acting at a point P and r is a displacement vector from a point Q to P as shown in Figure 8.16. Then, from (8.17), the torque about the point Q is given by . However, if the system is constrained so that it can only rotate about an axis OA, specified by the unit vector , as shown in Figure 8.16, then the relevant quantity is the component of in the direction of . This is called the torque about the axis . It is given by the triple scalar product and if we take to be along the z-axis, then

Moreover, the result is independent of the choice of the point Q on the axis of rotation and the location of P on the line of action of the force. To see this, suppose r′ is a vector from another point Q′ on OA intersecting the line of the force F through P at P′, as shown in Figure 8.16, Then,

(8.23)

But Q′Q is parallel to and PP′ is parallel to F, so these terms make no contribution to (8.23) and

(8.24)

which proves the result.

(ii) Triple vector product

Since (b × c) is perpendicular to both b and c and a × (b × c) is perpendicular to (b × c), it follows that the triple vector product a × (b × c) must be co-planar with b and c. Hence

(8.25)

where the unknown scalars α and β are conveniently determined by considering components, for example

To do this, we introduce

(8.26)

by (8.16). Using (8.16) again gives

on substituting from (8.26) and rearranging. Adding and subtracting a_xb_xc_x from this equation then gives

This can only hold for arbitrary vectors a, b, c if α = a · c and β = −a · b, so that (8.25) gives

(8.27)

Equation (8.27) not only facilitates the evaluation of triple vector products, but it is also extremely useful in deriving general results. For example, using it, one easily verifies that

(8.28)

while setting a = c in (8.27) gives

Rearranging the latter gives

(8.29)

which enables an arbitrary vector b to be expressed as a sum of vectors parallel to and perpendicular to a given vector a.

Finally, we consider triple vector products of the form (a × b) × c. Since vector products anti-commute, this is given by

(8.30)

On comparing (8.30) with (8.27), we see that

(8.31)

in general. In contrast to vector addition (8.1a) and (8.1b), vector products are not associative and in triple vector products the positions of the brackets are important.

A physical example of a triple vector product is the angular momentum of a particle of mass m fixed to the point P on a rigid body, as shown in Figure 8.12. By definition, the angular momentum L is given by

We have seen that the velocity is related to the angular velocity by , so L is given by the triple vector product

(8.32)

*8.2.4 Reciprocal vectors

In this section, we will use the properties of products of vectors described above to introduce the concept of reciprocal vectors. They enable the coefficients λ, μ, ν in the expansion (8.6a) of an arbitrary vector to be evaluated, as we shall show below. Reciprocal vectors play a central role in the theory of crystallography. This is because, by analogy with the expansion (8.6a), a crystalline substance may be described by a set of three non-coplanar lattice vectors a, b, c and the points on the lattice are given by the position vectors

where in this case λ, μ, ν are integers. For example, if the crystal is monatomic, the lattice points could correspond to the positions of the atoms. In addition, the reciprocal vectors describe X-ray diffraction patterns.⁴

To find λ, μ, ν for a set of non-coplanar vectors a, b, c, we define a set of corresponding reciprocal vectors a′, b′, c′ by

(8.34a)

and

(8.34b)

Assuming such vectors can be identified, then on multiplying both sides of (8.6a) by a′, we immediately obtain

by (8.34a) and (8.34b). Similar results applying for μ and ν, so that (8.6a) can be rewritten in the form

(8.35)

It remains to obtain explicit expressions for a′, b′, c′. To do this, we note that since a′ · b = a′ · c = 0, a′ must be perpendicular to both b and c and hence parallel or antiparallel to b × c. This implies that

where α is a constant. Substituting in the requirement a′a = 1 then gives α = [abc]^{− 1} and hence

(8.36a)

Similar arguments, together with (8.21), give

(8.36b)

where a · (b × c) ≠ 0, since the vectors are non-coplanar.

8.3.1 Straight lines

We will find the vector equation of a line through a given point A, with position vector a, in the direction of a given vector b, by reference to Figure 8.18. The position vector of any point P on the line is given by

where s is the length of the line AP in units of |b|. By allowing the scale parameter to vary over all values − ∞ < s < ∞, we obtain the position vectors for all points on the line. Thus, as the parameter s varies, the equation

(8.37a)

defines a straight line through the point A with position vector a in a direction of the vector b. It is often written in the standard form

(8.37b)

where λ = s|b| and is a unit vector in the b-direction as usual. More generally, one can choose any point A on the line and b can be replaced by any vector that is parallel or antiparallel to . For example, consider the equation for a straight line through the two points C and D, with vectors c and d, respectively. Then we could take A = C and use b = d − c to specify the direction of the line, so that (8.37) becomes

(8.38a)

or we could take A = D and use b = c − d, yielding

(8.38b)

These two equations (and others) describe the same straight line through C and D, although any given point on the line will correspond to a different value of s, depending on which form is used.

We turn now to Cartesian co-ordinates, in which

and

(8.39)

where cos α, cos β, cos γ are the direction cosines of . Substituting into (8.37a) then gives

in an obvious notation. Eliminating s gives the two equations

(8.40)

that are required to define a straight line in Cartesian co-ordinates in three dimensions.

As an example we will find the shortest distance from a point P to the line (8.37). This distance is the length d of the perpendicular from P to the line as shown in Figure 8.19a. If the position vector of P is p, then from the right-angled triangle,

(8.41a)

where is a unit vector in the direction of the line and we have used the definition of the vector product. We can also find the shortest distance d′ between two arbitrary lines in the directions of the vectors a and b. Referring to Figure 8.19b, the line normal to both a and b is a × b and so a unit vector normal to both lines is

Then if P is a point on the line in the direction a with position vector p and Q is a point on the line in the direction b with a position vector q, the line QP = (p − q) and the minimum distance between the lines is the component of QP along the unit normal , i.e.

(8.41b)

8.3.2 Planes

The above ideas can be extended to find the equation for a plane. Consider a plane defined by three non-collinear points A, B, C, with position vectors a, b, c. Then any other point R with position vector r that lies in the plane can be written in the form

(8.42)

as shown in Figure 8.20, where s and s′ are real parameters in the range − ∞ < s, s′ < ∞. This expression can then be re-arranged in the more symmetric form

(8.43a)

where α = 1 − s − s′, β = s, γ = s′ satisfy

(8.43b)

Equation (8.42) is similar in form to (8.38) used to describe a straight line, except that two parameters s and s′ are needed to define a given point on a plane, as opposed to one for a straight line. However, a very useful alternative description is obtained by considering a plane through the point A that is perpendicular to a given vector n. Then a vector r − a will be perpendicular to n if, and only if, the point r also lies in the plane. Hence

(8.44a)

is the condition that r lies in the plane and (8.44a) corresponds to the same plane as (8.42) if we choose n = (b × c) or n = −(b × c). In Cartesian co-ordinates, this becomes

(8.45a)

where α, β, γ are the direction ratios of n, i.e.

(8.45b)

and d = r · a. The latter parameter has a simple interpretation in the case that n is chosen to be a unit vector in the same direction as n. Then α, β, γ reduce to the direction cosines of (or n), satisfying [cf. Eq. (8.5)]

(8.46a)

and

(8.46b)

is the perpendicular distance from the origin to the plane, as can be seen from Figure 8.21a. The sign of d depends on which side of the plane the origin is situated relative to the direction of , where Figure 8.21a corresponds to the case d > 0.

To illustrate the use of (8.44), we will find the shortest distance from a point P, with position vector p, to the plane described by (8.44). Consider the vector (a − p), where a is the position vector of any point A in the plane, as shown in Figure 8.21b. Its component normal to the plane, which is the distance to the plane, is

(8.47)

where the sign will depend on the direction of (a − p) relative to . The same result can be extended to a line r = c + s b that is parallel to the plane, that is, for which . Then the perpendicular distance from a point on the line to the plane will be independent of which point is chosen and so will again be given by (8.47), where P is now any point on the line and A is any point in the plane. However, if the line and the plane are not parallel, that is, if , the line will pass through the plane and the minimum distance is zero.

We will conclude by considering the problem of two intersecting planes, as shown in Figure 8.22. If the vectors p and q are normal to the planes 1 and 2 respectively, then the angle θ between them is given by

(8.48)

which is also the angle of intersection of the two planes. For example, if the planes are

(8.49)

then we can take

giving θ = 99.6°. Finally, the line of intersection r of the two planes is contained in both planes, so that both equations (8.49) must be simultaneously satisfied, from which one obtains

where s is a scalar parameter. So the co-ordinates of any point on the line are x = s, y = 3s + 6, z = 8 − 5s and the vector equation of the line of intersection is