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Elements of Vector Algebra

Theorem 4.4.
−→
For any points A(x , y , z ) and B(x , y , z ) coordinates of vector AB can be found by
2
2
1
1
2
1
the following formulas: X = x − x ; Y = y − y ; Z = z − z . ⋆
1
2
2
2
1
1
z B
A

. . . . . . . . . A 1 B 1
O y

x

Figure 4.10 – Projection onto the coordinate axis

PROOF. Let’s show projections of points A and B for example, on axis Oy ( fig. 4.10). Coor-
−→
dinates of points A 1 and B 1 are y 1 and y 2 . According to the definition, Y = pr y AB = A 1 B 1 .
Therefore, A 1 B 1 = y 2 − y 1 and Y = y 2 − y 1 . In an analogy, the rest formulas can be proved. Thus,
−→
AB = (x 2 − x 1 ; y 2 − y 1 ; z 2 − z 1 ). 2

Remark 4.2. If a vector starts from the beginning of coordinates, i.e. x 1 = y 1 =
−−→
z 1 = 0, and x 2 = x, y 2 = y, z 2 = z, then coordinates X, Y, Z of vector OB coincide
with coordinates of its end: X = x, Y = y, Z = z. Such a vector is called as a
−−→
−→
radius-vector of point B and it is written as: OB = r = (x; y; z).

Remark 4.3. As a result of theorems about projections it is possible to get the
formulas of addition and multiplying of a vector by a digit in a coordinate view:

−→
−→
−→
− →
1. If a = (X 1 , Y 1 , Z 1 ) and b = (X 2 , Y 2 , Z 2 ), then a + b = (X 1 +Y 1 ; X 2 +Y 2 ; X 3 +Y 3 )
(in an analogy for the subtraction of vectors).
−→
−→
2. If a = (X 1 , Y 1 , Z 1 ), then for any real digit λ : λ · a = (λ · X 1 , λ · Y 1 , λ · Z 1 ).
From this formula, a condition of colinearity of two vectors in coordinates can
−→
−→
be written: X 2 = Y 2 = Z 2 . Indeed, the identity b = λ· a is equivalented to
X 1 Y 1 Z 1 −→
−→
equalities: X 2 = λ·X 1 ; Y 2 = λ·Y 1 ; Z 2 = λ·Z 1 . Thus, two vectors a and b are
colinear then and only then, when their coordinates are proportional.
4.6. Direction cosines of a vector

−→
Let’s consider a = (X 1 , Y 1 , Z 1 ) — any vector, which starts from the beginning of coordinates
and doesn’t belong to any of coordinate planes. Let’s draw through the end of this vector (point
A ) planes, that are perpendicular to coordinate axes. Obviously, we will get a rectangular
parallelepiped with diagonal OA (fig. 4.11).

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