Vector arithmetic

2.1.1 Vector addition

Vectors can be added one to the other, but the operation of ‘addition’ is not defined in the same way as it is for scalars, since the directions also need to be dealt with.

Vector addition is a graphical process: to add the vector bbb to the vector aaa, we slide the vector bbb until its butt-end touches the tip of aaa. The resultant vector,(aaa+bbb), which we shall call ccc, then begins at the butt-end of aaa and ends at the tip of bbb. Since both aaa and bbb can slide around without changing,

aaa+bbb=bbb+aaa=ccc

A vector can be made ‘negative’ by reversing its sense without changing its mag- nitude or direction; that is, swapping its butt-end for its tip without changing it in any other way. Vectors may then be subtracted from each other by recognizing

aaa−bbb=aaa+ (−bbb)

The negation, addition and subtraction of vectors is illustrated in Figure 2.1.

Fig. 2.1 Graphical illustration of vector addition.

2.1.2 Scalar multiplication

Because there is more information contained in a vector than in an ordinary number or scalar quantity, there are several ways to multiply vectors. The first and simplest

form of vector multiplication isscalar multiplication, or the multiplication of a vec- tor by a scalar. Because the scalar cannot contribute to the direction of the vector (since it has no direction of its own), it simply increases the magnitude (or length) of the vector without affecting its direction. For example, if ccc=3aaa, then the vector ccc is three times longer than aaa but points in the same direction. The scalar multiplication of a vector is illustrated in Figure 2.2.

Fig. 2.2 Graphical representation of the scalar product.

2.1.3 Dot product

The second form of vector multiplication is the dot product, which can only be performed between two vectors and results in a scalar. The dot product is denoted as aaa·bbb, and is defined asthe magnitude of the projection ofaaauponbbb, scaled by the magnitude ofbbb.

The vector aaa (the first argument of the dot product) can be expressed as the sum of any two vectors fff and ggg so that fff+ggg=aaa. While there are an infinite number of pairs of vectors fff and ggg which satisfy the requirement that fff+ggg=aaa, there is only one pair of fff and ggg which can be chosen so that (i) fff lies parallel with bbb (which can be indicated fffbbb), and (ii) ggg is perpendicular to bbb (which can be indicated ggg⊥bbb). For this unique fff and ggg, then, aaa·bbb=|bbb||fff|; in other words, the result of the dot product is the scalar length of the vector bbb multiplied by the scalar length of the vector fff lying parallel with bbb which simultaneously satisfies fff+ggg=aaa and fff⊥ggg. This property is illustrated in Figure 2.3.

From trigonometry, we recognize that the definition of the dot product may be much more simply expressed as

aaa·bbb=|aaa||bbb|cos(θ)

where θ is the angle subtended between aaa and bbb. Since|aaa|,|bbb|andθ ^{do not de-} pend on direction, it follows that(aaa·bbb) = (bbb·aaa). Also, ifθ=90^◦(i.e. aaa and bbb are perpendicular), then aaa·bbb=0.

Fig. 2.3 Graphical representation of the dot product; aaa·bbb=|fff||bbb|=|aaa||bbb|cos(θ)

2.1.4 Cross-product

The third form of vector multiplication is thecross product. The cross product can also only be performed between two vectors, but results in another vector. The cross product of the vectors aaa and bbb is denoted as aaa×bbb. The magnitude and direction of the cross product are defined separately.

For the magnitude of the cross-product, we express the vector aaa (the first ar- gument of the cross product) as the vector sum of any two vectors fff111 and fff222 so that aaa=fff111+fff222. Again, an infinite number of pairs of vectors fff111and fff222will satisfy aaa=fff111+fff222, but only one pair exist such that (i) fff111bbb and (ii) fff222⊥bbb, as illustrated in Figure 2.4. Themagnitudeof ggg=aaa×bbb, then, is the scalar length of the vector b

b

b multiplied by the scalar length of the vector fff222 lying perpendicular to bbb which simultaneously satisfies aaa=fff111+fff222and fff222⊥fff111.

Fig. 2.4 Graphical representation of the magnitude of the cross product; |aaa×bbb|=|fff₂₂₂||bbb|=

|aaa||bbb|sin(θ)

From trigonometry, as before, we recognize that the magnitude of the cross prod- uct may be much more easily described as

|ggg|=|aaa×bbb|=|aaa||bbb|sin(θ) whereθis again the angle subtended between aaa and bbb.

The direction of the cross product ggg=aaa×bbb is defined as being mutually perpen- dicular to aaa and bbb. This means that if a plane were defined such that both aaa and bbb lay in that plane, ggg would be normal (or perpendicular) to that plane. The cross-product is therefore inherently a three-dimensional operator. By convention, the sense of ggg is dictated by theright-hand rule: if you align the fingers of your right hand with aaa and then bend your fingers so that they are aligned with bbb (so that your fingers have swept out the angleθ), your thumb will point in the direction of aaa×bbb. It stands to reason, then, that

aaa×bbb=−bbb×aaa The cross-product is illustrated graphically in Figure 2.5.

Fig. 2.5 Graphical representation of the cross product; ggg=aaa×bbb

While the cross-product initially seems to be an obscure construct with no ap- plication in the physical world, it is extremely useful in that it is a mathematical tool which allows us to simply and efficiently deal with things which are rotating in three-dimensional space.

The equations listed below are useful trigonometric vector identities, given the vectors aaa, bbb and ccc, and the scalarξ^.

ξ(aaa+bbb) =ξ^aaa+ξ^bbb aaa·bbb=|aaa||bbb|cos(θ)

|aaa×bbb|=|aaa||bbb|sin(θ) aaa·bbb=bbb·aaa aaa×bbb=−bbb×aaa aaa×aaa=0

aaa·aaa=|aaa|² ξ^aaa·bbb=ξ(aaa·bbb) ξ^aaa×bbb=ξ(aaa×bbb) (aaa+bbb)·ccc=aaa·ccc+bbb·ccc aaa×(bbb+ccc) = (aaa×bbb) + (aaa×ccc)

aaa×bbb×ccc= (aaa·ccc)bbb−(aaa·bbb)ccc