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Projections of a vector onto the coordinate axis

Theorem 4.2.
−→
The projection of a sum of two vectors a and b on axis l equals to the sum of projections of
−→
these vectors on the same axis. ⋆

−→ −→ −−→
PROOF. Let’s consider AC = AB +BC ( fig. 4.9), x 1 , x 2 , x 3 — coordinates of projections A l ,
−→ −−→ −→
B l ,C l onaxisl pointsA,B,C.Thenpr−→AB = x 2 −x 1 ;pr−→BC = x 3 −x 2 ;pr−→AC = x 3 −x 1 .
l l l
−→ −→ −−→
But x 3 − x 1 = (x 2 − x 1 ) + (x 3 − x 2 ), so pr−→AC = pr−→AB + pr−→BC. 2
l l l
This theorem spreads up to any amount of elements.

B

C
A

O . . . . . . . . . l
A 1 B 1 C 1
Figure 4.9 – Projection of a sum

Theorem 4.3.
On multiplying vector projection a into any digit λ, its projection on axis l multiplies into that
−→
−→
−→
digit: pr−→(λ · a ) = |λ| · pr−→ a .
l l

−→
−→
PROOF. Let’s note, when λ > 0, then λ a ⇈ a , i.e. forms the same angle with axis l; when
−→
−→
λ < 0, then λ a ↑↓ a , i.e. forms with axisl angle π − φ. Using theorem 4.5, we will get:
−→
−→
−→
− →
−→
λ > 0 : pr−→(λ · a ) = |λ · a | · cos φ = |λ| · | a | cos φ = λ · | a | cos φ = λ · pr−→ a ,
l l
−→
−→
−→
λ < 0 : pr−→(λ · a ) = |λ · a | · cos(π − φ) = |λ| · | a | cos(π − φ) =
l
−→
−→
−→
= −λ · | a | · (− cos φ) = λ · | a | · cos φ = λ · pr−→ a . 2
l
Remark 4.1. The projection of a subtraction of two vectors on axis l equals to
the subtraction of projections these vectors on the same axis.
4.5. Projections of a vector onto the coordinate axis

−→
Let’s consider a rectangular system of coordinates Oxyz and an arbitrary vector AB, X =
−→ −→ −→
pr x AB, Y = pr y AB, Z = pr z AB. These projections on coordinate axis are called as coordi-
−→ −→
nates of vector AB. It can be denoted by: AB = (X; Y ; Z).

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