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Elements of Vector Algebra

− → −→ −→
−→ −→
k =1. These a vectors are called as orths. Due to orths i , j , k are non-complanar, these
vectors form the basis, which is called a rectangular (or orthogonal) basis.
− →
Let’s consider some vector a in space, and move it parallel to itself in order its beginning
−−→ −→
equals to the origin: OM = a .

M z

M(x, y, z)

−→
−→ z k
k −→
j
O . . . . . . . . . . . . . M y
−→
−→ i
x i
M x
−→ P
y j

Figure 4.13 – Expansion of a vector by basis

−−→
Let’s build a parallelepiped with a diagonal that equals to vector OM. It is obvious, that
−−→ −−→ −−−→ −→ −−−→ −−→ −−→ −−→ −−→
−→ −→
a = OM = OM x + M x P + PM. Due to M x P = OM y , PM = OM z , then a = OM x +
−−→ −−→ −−→ −−→ −−→ − →
OM y + OM z . Vectors OM x , OM y , OM z are component parts of vector a on axis Ox, Oy,
−−→ −−→ −→ −−→ −−→ −→ −−→ −−→ −→
Oz, because OM x = pr x OM · i , OM y =pr y OM · j , OM z = pr z OM · k . Having marked
−−→
projections of vector OM on axis, correspondingly, as X, Y, Z, we will get the expansion of
−→ −→ −→ −→ −→ −→ −→
−→
vector a through basis i , j , k : a = X · i +Y · j +Z · k . Projections X, Y, Z are called
−→
as the rectangular coordinates of vector a .

4.9. Separation of a closed interval in a given ratio

To separate closed interval M 1 M 2 in the given ratio λ > 0 means to find on that interval such
point M, for which the equality M 1 M = λ or M 1 M = λ·MM 2 takes place. Suppose, coordinates
MM 2
of points M 1 and M 2 are: M 1 = (x 1 ; y 1 ; z 1 ), M 2 = (x 2 ; y 2 ; z 2 ). Let’s find coordinates of point
−−−→ −→
M. Obviously, M 1 M = λ · MM 2 or

− → − → −→ −→ −→ − →
(x − x 1 ) · i + (y − y 1 ) · j + (z − z 1 ) · k = λ · [(x 2 − x) · i + (y 2 − y) · j + (z 2 − z) · k ]

Following the equality of vectors, we will get equalities of their projections: (x − x 1 ) =
λ · (x 2 − x); (y − y 1 ) = λ · (y 2 − y); (z − z 1 ) = λ · (z 2 − z). Thus,

x 1 + λ · x 2 y 1 + λ · y 2 z 1 + λ · z 2
x = , y = , z = .
1 + λ 1 + λ 1 + λ
If point M is on the middle of interval M 1 M 2 , then M 1 M = MM 2 , λ = 1. In this case coordi-
nates of point M are: x = x 1 +x 2 , y = y 1 +y 2 , z = z 1 +z 2 . So, every coordinate of the middle of
2 2 2
the interval equals to the average arithmetic of corresponding coordinates of its ends.

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