Vector products

We distinguish three different product of vectors named according to the re-spective result:

• Scalar product

• Vector product

• Dyadic product.

Scalar product: The scalar (dot) product A· B of two vectors A and B is denoted by a dot and it can be intuitively illustrated. Figure 2.9 depicts a vector A being projected onto a unit vector ˆe with the result

A· ˆe = A cos φ, (2.12)

if φ is the angle between A and ˆe. Replacing ˆe by a vector B with magni-tude B, the generalization of (2.12) reads as

A· B = B · A

= AB cosφ (2.13)

and defines the (commutative) scalar product A· B.

A

ˆe

A cos φ = A · eˆ φ

FIGURE 2.9

Illustration of the scalar product.

We obviously have A· B = 0 if A and B are orthogonal to each other;

consequently, the orthogonality of two vectors is guaranteed finding the value zero of their scalar product.

The orthonormal trihedron of cartesian coordinates has the property:

e_x· ey= 0,

e_x· ez= 0, (2.14)

e_y· ez= 0;

e_x· ex= 1,

e_y· ey= 1, (2.15)

e_z· ez= 1.

Numbering cartesian coordinates according to x_i, i = 1, 2, 3, with the trihe-dron e_xi, i = 1, 2, 3, and utilizing the Kronecker symbol

δij =

1 for i = j

0 for i
= j, (2.16)

we can write the six equations of (2.14) and (2.15) as a single equation:

e_xi· exj =δij for i, j = 1, 2, 3. (2.17) The scalar product is useful to calculate the components of a vector A, for example, in Cartesian coordinates; with (2.12), it follows per definition

A_x= A· ex,

Ay= A· ey, (2.18)

A_z= A· ez.

Now we calculate

A· B = (Axe_x+ A_ye_y+ A_ze_z)· (Bxe_x+ B_ye_y+ B_ze_z) (2.19) with the (Cartesian) component representation of A and B and formally find by distributive multiplication and utilization of (2.17) observing the commu-tative property of the scalar product

A· B = AxBx+ AyBy+ AzBz. (2.20) That way, we have the possibility to find the value of the scalar product if Cartesian components of the respective vectors are given. Similarly, the angle between two vectors with nonzero magnitudes is obtained as

cosφ = A· B AB

= AxBx+ AyBy+ AzBz



A²_x+ A²_y+ A²_z



B²_x+ B_y²+ B_z²

. (2.21)

The square root of the scalar product A· A obviously yields the magni-tude of A:

A =
A· A

=



A²_x+ A²_y+ A²_z; (2.22) in addition, we obtain

Aˆ = A A

= A

√A· A

= A_x

A e_x+A_y

A e_y+A_z

A e_z (2.23)

as the unit vector ˆA in the direction of A. If applied to the vector of position, (2.23) provides

Rˆ = x

x²+ y²+ z²e_x+ y

x²+ y²+ z²e_y+ z

x²+ y²+ z²e_z. (2.24)

We quote another two—abbreviated—notations for the scalar product.

The serially numbered version of (2.18)

Ax_i = A· exi for i = 1, 2, 3 (2.25) and equally for B results in

A· B =

3 i=1

Ax_iBx_i (2.26)

instead of (2.19). If we agree that the xi—as in this case—are cartesian co-ordinates, we can continue, according to Ax_i=⇒ Ai, Bx_i=⇒ Bi, abbreviat-ing (2.26):

A· B =

3 i=1

A_iB_i. (2.27)

Einstein’s summation convention goes even further omitting the summation sign in (2.27):

A· B = AiBi. (2.28)

Equation 2.28 is translated as: If an index on one side of an equation—in this case i—appears at least twice and is not found on the other side, a summa-tion from i = 1 to i = 3 is understood, the index is contracted;⁶ if the index also appears on the other side, it is not contracted. This summation conven-tion is extensively applied in the literature on elastodynamics (e.g., Achenbach 1973; de Hoop 1995); nevertheless, we generally prefer the coordinate-free rep-resentation A· B instead of (2.28), because it is much more practical for ana-lytical derivations; yet, in case numbers are requested as a result of a physical problem, one must rely on coordinates.

Once again, we consider a specimen as in Figure 2.6 and imagine that a point-like piezoelectric “transducer” at the measurement point PM(R) exclu-sively measures the component of the particle displacement u(R, t) normal to the surface (Figure 2.10). To characterize this “normal component” un(R, t)

P_M(R)

O R

u_n(R, t) u(R, t)

n

FIGURE 2.10

Normal component of the particle displacement.

6Therefore, a dot product (scalar product) A· B implies contraction of adjacent indices of the scalar components of the vectors in the immediate neighborhood of the dot.

A

B F = C

C = A × B

π2

π2 φ

FIGURE 2.11

Definition of the vector product.

mathematically, we define a unit vector n being orthogonal to the surface of the specimen.⁷ Per definition, we have

un(R, t) = u(R, t)· n = n · u(R, t). (2.29) Being difficult to simultaneously measure the tangential components of u(R, t), the normal component un(R, t) for “all” points R on a measurement surface SM and all times t is generally the maximum obtainable informa-tion in US-NDT. In connecinforma-tion with imaging methods, we will learn how to process it.

Vector product: The definition of the vector product A× B—that is, Across B—is illustrated in Figure 2.11. Two vectors A and B span a rhom-boid with the area

F = AB sinφ; (2.30)

the vector C with magnitude F being right-handed⁸ orthogonal to the rhom-boid area is called the vector product

C = A× B (2.31)

of A and B. Because of its definition implying right-handedness, the vector product is not commutative; we rather have⁹

B× A = −A × B. (2.32)

7This unit vector multiply appears with the same meaning, hence, the hat is omitted.

To calculate it, the surface must be suitably parameterized.

8The orthogonality of C to the rhomboid area only defines the shaft of the arrow rep-resenting C. With regard to the tip, there is the choice “upward” or “downward.” The arbitrary decision is “up” specified by the right-hand rule: If the cranked fingers of the right hand point from A to B, the vector product C = A× B should point into the direction of the thumb of the right hand. Because of this choice, the vector product yields a so-called axial or pseudo-vector.

9Compare Footnote 8.

Obviously, two vectors are parallel or antiparallel if their cross product vanishes. It follows:

e_x× ex= 0,

e_y× ey= 0, (2.33)

e_z× ez= 0.

The symbol 0 denotes the null vector, that is to say a vector with zero cartesian scalar components. We immediately verify

e_y× ez= e_x,

e_x× ey = e_z, (2.34)

e_z× ex= e_y.

Distributive multiplication of the component representations of A and B uti-lizing (2.33) and (2.34) yields

C= (Axe_x+ Aye_y+ Aze_z)× (Bxe_x+ Bye_y+ Bze_z)

= (A_yB_z− AzB_y) e_x+ (A_zB_x− AxB_z) e_y+ (A_xB_y− AyB_x) e_z (2.35) for the components of C.

Orthogonality of the cross product to its vector factors has as a consequence

A· (A × B) = 0,

B· (A × B) = 0. (2.36)

The product

A· (B × C) = C · (A × B) = B · (C × A) (2.37) is nothing but the volume of the parallelepiped spanned by A, B, C.

The relation

n× u(R, t) = utan(R, t) (2.38) defines the vector of the particle displacement tangential to the surface being characterized by the normal vector n, i.e., its tangential “component.” For instance, electromagnetic ultrasonic transducers or laser vibrometers are able to measure this particular component. Note: The vector tangential compo-nent u_tan(R, t) is orthogonal to n and u(R, t), it is not in the plane spanned by n and u(R, t) as it is true for the vector tangential component u_t(R, t) (Equation 2.97; Figure 2.12).

Dyadic product: Now, we define a dyadic product of two vectors where the intuitive interpretation only follows after its definition and application, hence

we proceed formally and put two vectors adjacent to each other without dot or cross in terms of their cartesian component representation:

A B = (A_xe_x+ A_ye_y+ A_ze_z)(B_xe_x+ B_ye_y+ B_ze_z). (2.39) Distributive multiplication produces the pertinent dyadic products of the unit vectors:

A B = AxBxe_xe_x+ AxBye_xe_y+ AxBze_xe_z + A_yB_xe_ye_x+ A_yB_ye_ye_y+ A_yB_ze_ye_z

+ AzBxe_ze_x+ AzBye_ze_y+ AzBze_ze_z (2.40)

=

3 i=1

3 j=1

AxiBxje_xie_xj (2.41)

= Ax_iBx_je_xie_xj (summation convention). (2.42) Summation convention means that summation from 1 to 3 on the right-hand side affects the indices i and j appearing twice on that side.

The vector with the component representation

A = A_xe_x+ A_ye_y+ A_ze_z (2.43) can be written as a single-column matrix (column vector)

A =

⎛

⎝Ax

Ay

Az

⎞

⎠ (2.44)

or as a single-row matrix (row vector) A^T=

A_x A_y A_z

, (2.45)

being the transpose—indicated by the upper index T—of the single-column matrix.¹⁰The unit vectors in (2.43) refer to the position of the scalar compo-nent in the perticompo-nent matrix scheme. Similarly, we can choose the scheme

A B=

⎛

⎝AxBx AxBy AxBz

AyBx AyBy AyBz

AzBx AzBy AzBz

⎞

⎠ (2.46)

of a 3×3-matrix for the dyadic product (2.40)—the dyadic A B. Obviously, the dyadic products e_xie_xj, i, j = 1, 2, 3, indicate the position of the element Ax_iBx_j in the matrix if we agree upon the choice of the first index as row index and the second index as column index. We adhere that in this sense a dyadic possesses nine scalar components in contrast to the three scalar

10We only must know the coordinate system for the components.

components of a vector; nevertheless, in the present case, the nine components are determined by the six vector components of the two vectors forming the dyadic product. From the definition of the dyadic product, we deduce that it is not commutative:

A B
= B A. (2.47)

The dyadic product yields a descriptive meaning when applied via a dot product (contraction) from left or right to a vector. Hence, we try to interpret the operation

A B· C (2.48)

or

C· A B (2.49)

writing A B· C in components

A B· C =

and using (2.17) to calculate

A B· C =

The left-sided contraction of a dyadic product with a vector is nothing but the contraction of the indices of the adjacent vectors—in this case, B and C; the scalar product B· C shows up as a scalar factor of the remaining vector A, the left factor of the dyadic product.

In complete analogy, we compute

C· A B = (C · A)B, (2.52)

and obviously we find

A B· C
= C · A B. (2.53)

The dyadic operator A B rotates the vector C into the direction of the vec-tor A according to A B· C and the vector C into the direction of the vector B according to C· A B.

Commercially available shear wave transducers radiate transverse waves under various angles applying normal forces to surfaces: The related particle displacement as a vector has quite different directions that do not comply with the normal to the surface. Therefore, the transformation force =⇒ wave must be mathematically procured by a dyadic operator; in the case of point-like forces, it is just Green’s dyadic. Its explicit mathematical structure is required to model sound fields of piezoelectric transducers.

Utilizing the matrix representations (2.46) and (2.44) of A B and C, we find A B· C as a single-column matrix resulting from matrix multiplication:

⎛

Analogously, we find C· A B as a single-row matrix:

(C_x, C_y, C_z)

The explicit calculation of A B· C (or C · A B) becomes most obvious utilizing the summation convention

We nicely see that the dot product contracts adjacent indices, i.e., one index—

in this case, j—disappears.

It is evident that

A B× C (2.57)

and

C× A B (2.58)

become meaningful through (2.51): A B× C is the dyadic (!) product of the vector A with the (axial) vector B× C, and C × A B is the dyadic D B mit D = C× A.

Linear independence: Three vectors A₁, A₂, A₃are linearly independent if α1A₁+α2A₂+α3A₃= 0 (2.59) only holds forα1=α2=α3= 0. Therefore, linear dependence implies that the three vectors span a triangle.

Complex valued vectors: The frequency spectrum¹¹ u(R,ω) of the time-dependent particle displacement u(R, t) apparently is a vector field

u(R,ω) = ux(R,ω) ex+ uy(R,ω) ey+ uz(R,ω) ez, (2.60) whose components are frequency spectra of the components of u(R, t) (Equa-tion 2.10). Yet, frequency spectra generally are complex valued func(Equa-tions of the (real) variableω (Section 2.3) with consequences regarding algebraic oper-ations like, for instance, computing the magnitude of u(R,ω). If we calculate u(R,ω) · u(R, ω) = u²x(R,ω) + u²y(R,ω) + u²z(R,ω), (2.61) the single terms

u²_xi(R,ω) = u²x_iR(R,ω) − u²x_iI(R,ω) + 2jux_iR(R,ω)ux_iI(R,ω),

i = 1, 2, 3, (2.62)

are complex numbers with real

u_xiR(R,ω) = {uxi(R,ω)}, i = 1, 2, 3, (2.63) and imaginary part

ux_iI(R,ω) = {ux_i(R,ω)}, i = 1, 2, 3, (2.64) of ux_i(R,ω). As a consequence, (2.61) is no longer the square of the “length”

of the complex valued vector u(R,ω). However, if we investigate the so-called Hermite product

u(R,ω) · u^∗(R,ω) = |ux(R,ω)|²+|uy(R,ω)|²+|uz(R,ω)|², (2.65)

11For physical quantities, we use the same character u for the (spatially dependent) time function u(R, t) and for the (spatially dependent) spectrum u(R, ω) and distinguish them through explicit indication of the variable t or ω, respectively; often one finds ˆu, ˜u, ¯u, U for the spectrum. Note that the physical dimension of u(R, ω) is equal to the physical dimension of u(R, t) multiplied by the physical dimension “time.”

where u^∗(R,ω) has the complex conjugate components of u(R, ω), then the magnitudes of the complex numbers ux_i(R,ω) appearing in (2.64)

|uxi(R,ω)| =

{uxi(R,ω)}²+{uxi(R,ω)}², i = 1, 2, 3, (2.66) are real valued.

Generalizing (2.22), we define the real positive length of a complex vector C according to

|C| =

C· C^∗. (2.67)

In document Karl-Jorg Langenberg, Rene Marklein, Klaus Mayer. Ultrasonic Nondestructive Testing of Materials - Theoretical Foundations 2012. (Page 38-48)