Page 23 - 4263

Projection of a vector onto the axis

−→ −→
a a
− →
− → − →
c = a + b
− →
b
−→
b

. . . . . .

Figure 4.5 – Parallelogram rule

− →
a
−→
−→
a − b
−b

Figure 4.6 – Subtracting vectors
. . . .

4.3.2. Multiplying of a vector by a digit

−→
−→
Definition 4.3. A multiple of vector a by a real digit λ is called such vector b ,

−→
−→
with the module equals to |λ| · b , which has the same direction with vector a
when λ > 0 and the opposite direction when λ < 0, and is a null-vector when λ = 0.
− →
− →
This operation can be written as: b = λ · a . Thus, target vector b is colinear to
−→
the given vector a , moreover, the last one is stretched by λ times when |λ| > 1 and
−→
is compressed in 1/λ times when |λ| < 1. ✓
Let’s write the main properties of linear operations:
−→ −→
−→
− →
1. a + b = b + a (commutative property)
− → − →
−→
−→
−→
− →
2. ( a + b ) + c = a + ( b + c ) (associative property for the sum of vectors)
−→ −→
3. λ · (µ · b ) = (λ · µ) · b (associative property for the multiple of vectors)
− → −→ −→
4. (λ + µ) · b = λ · b + µ · b (distributive property for the sum of numbers)
−→ − →
−→
−→
5. λ · ( a + b ) = λ · a + λ · b (distributive property for the sum of vectors),
− →
−→
where λ and µ are any real numbers, a and b — any vectors. Proof of
these properties follows from the definitions of linear operations.
4.4. Projection of a vector onto the axis
−→
Suppose l — is any axis, AB — vector in space. Let’s mark as A 1 and B 1 projections of the
beginning (point A) and the end (point B) of this vector (fig. 4.7) a. Suppose x 1 is the coordinate
of point A and x 2 is the coordinate of point B on axis l.

Definition 4.4. A subtraction x 2 − x 1 between coordinates of projections of the
−→
−→
end and beginning of vector AB into axis l is called as projection of vector AB into

23