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Linear dependence of vectors. Basis

It is known that the square of a length of a diagonal of the rectangular parallelepiped equals
2 2 2 2
to the sum of squares of lengths of three its dimensions. So, |OA| = |OA x | +|OA y | +|OA z |
−→ 2
2
2
2
or | a | = X + Y + Z . Therefore, the module of a vector:
√
−→
2
2
2
| a | = X + Y + Z . (4.2)
−→
Let’s mark α, β, γ — angles between vector a and coordinates axes.
z
A z

a z

A
−
γ →
a
O . . . . . . . . . . . . . . . . β a y A y y
α

a x

x
A x
Figure 4.11 – Direction cosines

According to formulas (4.1) and (4.2):

X X
cos α = −→ = √ ,
2
2
| a | X + Y + Z 2
Y Y
cos β = −→ = √ ,
2
2
| a | X + Y + Z 2
Z Z
cos γ = −→ = √ (4.3)
2
2
| a | X + Y + Z 2
cos α, cos β, cos γ are called direction cosines of vector a , angles α, β, γ — direction (direc-
− →
− →
tional) angles of vector a . Raising to the square power both parts of the last formulas, and
having added term, we will get a formula, which is used for any vector:
2
2
2
cos α + cos β + cos γ = 1. (4.4)
4.7. Linear dependence of vectors. Basis

Definition 4.5. Vectors a 1 , a 2 , . . . , a n are called linear dependent, if there is
−→
−→ −→
a set of digits λ 1 , λ 2 , . . . , λ n (not all of them are equal to zero), that the equality
takes place:
−→
−→
−→
λ 1 · a 1 + λ 2 · a 2 + λ n · a n = 0 (4.5)
Vectors a 1 , a 2 , . . . , a n are called linear independent, if equality (4.5) takes
−→
−→ −→
place when λ 1 = λ 2 = . . . = λ n = 0. ✓

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