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A rectangular basis. Expansion of a vector by basis

PROOF. Proof of this theorem can be done in the same way as Theorem 4.5. The above explained
twocasesmustbetakenintoconsideration: 1)thereisatripleofcomplanarvectorsamongthem(com-
planar vectors are the ones that belong to the same plane); 2) there isn’t any triple of complanar vec-
tors among given ones. 2

There are some conclusions to be made from this theorem:
1. If the number of given vectors in space is more than four, then these
vectors are linear dependent.
2. For three vectors in space to be complanar it is necessary and sufficient
to be linear dependent.
3. For three vectors in space to be linear independent it is necessary and
sufficient to be non-complanar.
As follows from these conclusions, the maximum number of linear independent vectors in space
equals to three.
Let’s introduce a concept of a basis in a plane.

Definition 4.6. Any two linear independent (non - colinear) vectors in a plane
are called as basis on a plane. Obviously, there can be many basis on a plane, and
despite base vectors, any third vector on a plane can be expanded by these basis
vectors. ✓

Theorem 4.8.
−→
−→
−→
The expansion of vector a by basis vectors b and c is unique. ⋆
−→
−→
−→
PROOF. Let’s assume that, besides from expansion a = λ 1 · b + λ 2 · c , there is another
−→
−→
−→
expansion a = µ 1 · b + µ 2 · c . We are going to prove that µ 1 = λ 1 ; µ 2 = λ 2 . Indeed, the
−→ −→
−→
subtraction of the above expansions is: 0 = (λ 1 − µ 1 ) · b + (λ 2 − µ 2 ) · c . But basis vectors b
−→
and c are linear independent, so (λ 1 − µ 1 ) = 0 and (λ 2 − µ 2 ) = 0. Thus µ 1 = λ 1 ; µ 2 = λ 2 , i.e.
− →
− →
−→
the expansion of vector a by basis vectors b and c is unique, moreover digits λ 1 and λ 2 are called
−→ −→
− →
as coordinates of vector a in basis b and c . 2
Definition 4.7. Any three linear independent (non-complanar) vectors are
− →
called as basis in space. In an analogy to a plane, any vector d can be ex-
−→
−→
−→
−→
panded by basis vectors a , b and c , moreover, this expansion is unique: d =
−→
− →
−→
−→
λ 1 · a + λ 2 · b + λ 3 · c . Digits λ 1 , λ 2 , λ 3 are called coordinates of vector d in basis
−→ −→ −→ ✓
a , b , c .
4.8. A rectangular basis. Expansion of a vector by basis
Let’s consider a rectangular system of coordinates in space Oxyz (fig. 4.13). Let’s choose single
vectors on each coordinate axis with directions equal to positive directions of those axis. On

−→ − → − → −→
−→
axis Ox we will get vector i , on axis Oy — vector j , on axis Oz — vector k : i = j =
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