Mathematical Physics Vol 1

3. Examples

3.1 Vector algebra

Problem2

Prove: a) commutativity of the sum of vectors. b) associativity of the sum of vectors.

Proof a) Consider two vectors a and b (Fig. 3.1). The common point B is the point where vector a ends and vector b begins.

Using the parallelogram rule for the sum of vectors, vector c is ob tained, which starts at point A , and ends at point C , c = a + b . Let us now translate the vector b so that it starts in point A andvector a so that it starts in point D , which is the end point of vector b .

Figure 3.1: Figure in Problem 2.

It is obvious from the figure that −→ AB + −→ BC = −→ AC and −→ AD + −→ DC = −→ AC , that is, a + b = c and b + a = c . It follows that a + b = b + a , namely that the sum of vectors is a commutative operation. The figure depicts a parallelogram ABCD with the diagonal AC , which is why the vector addition rule is called the parallelogram rule.