Mathematical Physics Vol 1

Chapter 1. Vector algebra

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Figure 1.6: Force displacement - deformable body

For example, in Fig. 1.6 a the rod is strained to tension, and the rod elongates. If the points of application of both forces move, say, to the center of the rod, Fig. 1.6 b , the rod will not be strained. Finally, if forces are applied to opposite ends of the rod, then the rod will be strained to pressure (Fig. 1.6 c ), and the rod is shortened. Thus, from the standpoint of movement or resting of the rod, it is completely irrelevant whether it is affected by forces as in Figures 1.6. All three cases are equivalent. But from the standpoint of determining the internal forces in individual sections of the rod, the difference is essential.

The following vectors can be distinguished: - free (they move parallel to themselves, but do not change; an example for this type of vector is the coupling momentum, the translation vector), - sliding or vector bound to a line (it does not change when moving along the carrier line;

for example, the force acting on a rigid body) and - bound to a point (for example, volume forces).

R Note that the operations to be defined will only apply to free vectors, unless otherwise noted.

Starting from the idea of vectors as point displacements, we conclude that two vectors are equal if the oriented segments representing them are equal in length (equal in magnitude), and their directions are the same. We will denote this by

a=b .

(1.6)

Fig. 1.7 shows vector pairs that are not equal because they differ in magnitude (Fig. 1.7(a)) or direction (Fig. 1.7(b) and 1.7(c)).

a

b

b

a

b

a

(a)

(b)

(c)

Figure 1.7: Vectors that differ (a) in magnitude (b) and (c) in direction.

We will denote the length (magnitude) of the vector a by | a | or shortly a .