Mathematical Physics Vol 1

1.4 Operations on vectors

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Figure 1.4: Sum of vectors as equilibrium of forces.

In literature, this rule is known as the parallelogram law of addition, as (see Figures 1.3 and 1.4) the sum of vectors a and b is represented by the diagonal of the parallelogram ABCD . Addition of vectors is, thus, a binary operation over a set of vectors V , by which a vector c is uniquely assigned to vectors a , b ∈ V . The fact that many quantities in physics can be represented by oriented straight line segments, which are summed according to the parallelograms law, prompts the study of vectors in more depth. Thus, by introducing vectors, physical quantities are geometrized.

R Note that there are situations in physics in which it is necessary to impose a boundary on the start point or position of the line - carrier of the observed vector. Two examples ( rigid and deformable body ) 6 follow.

Example 1 Let us observe a rigidbody . One of the axioms of statics is: two systems of forces are statically equivalent if the difference between them amounts to a system of forces in equilibrium.

Figure 1.5: Movement of the force - rigid body

An important consequence of this axiom is: the point of application of the force on a rigid body can move along the line of action of the force. Namely, if a system in equilibrium (⃗ S ′ ,⃗ S ) is added in point B (on the line of action of the force) (Fig. 1.5), and then the system in equilibrium (⃗ S ′ − point of application B ,⃗ S − point of application A ) is removed, then the force⃗ S still remains, but with the point of application B . However, if the body is viewed as deformable, it is irrelevant at which point of the body the force is going to be applied.

6 A body in which the distance between any two points does not change during its movement.