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Elements of Vector Algebra

4.3.1. Addition of vectors

−→ −→ −→ −→ −→ −→
Let’s consider a few vectors ( a 1 , a 2 , a 3 , a 4 , a 5 , a 6 ), located as it shown on (fig. 4.3) (the
beginning of the second vector coincides with the end of the first one, etc). It is said that such
vectors are located consistently.

Definition 4.2. A vector, which connects the beginning of the first vector with
the end of the last one, is called as a sum of n vectors. Here a polygonal appears,
and so, the rule of adding vectors, is called the rule of polygons. ✓

−→
− →
−→
−→
−→
−→
−→
Example 4.1. a 1 + a 2 + a 3 + a 4 + a 5 + a 6 = a (fig. 4.3). ,
−→
−→
Particularly, two vectors a , b can be added according to the triangle rule (fig. 4.4).
−→
a 2
−→
a 1
−→
. . . . . . . . a 3
− →
a
−→ −→ −→
a 6 a 5 a 4
Figure 4.3 – Addition of vectors

−→ −→
a a
−→
b
−
→
−
→
−
→
c = a + b
−→
b
. . . . . .
Figure 4.4 – Triangle rule

− →
To do so, it is enough to move vector b parallel to itself in order its beginning be equal to
−→
−→
−→
the end of vector a and connects the beginning of vector a with the end of vector b .
−→
−→
Two non-colinear vectors a , b can be added in another way — according to a parallelogram
rule. To do so, it is enough to move these vectors in any point O of space parallel to itself and
build a parallelogram based on these vectors. Therefore, a vector which serves as a diagonal of
− →
−→
a parallelogram with its beginning in point O, equals to the sum of vectors a and b .
−→
− →
According to (fig. 4.5), the sum of two vectors has a commutative property: a + b =
−→ −→ −→ −→ −→
b + a . Subtraction of vectors a and b can be led to addition of vector a and the opposite
− → −→ −→ −→ −→
vector to vector b . Thus, a − b = a + (−1) · b .
−→
−→
To do so, move vectors a and (− b ) parallel to itself in any point of space, build a par-
allelogram based on these vectors and show a vector-diagonal, which runs from point O (fig.
4.6).
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