Linear Operations on Vectors
Definition. The product of a vector 
and a real number l is the vector 
defined by the following conditions:
(1) 
;
(2) The vector 
is collinear to ;
3. The vectors 
and have the same direction if λ>0 and opposite directions if λ<0. If λ=0, then the direction of the vectors is arbitrary.
3
Property 1.For any numbers α and β and any vector 
,
α(β
)= (αβ) .
The vectors on both sides have the same length 
; moreover they are collinear and have the same direction. Their direction coincides with the direction of if α and β are of the same sign and is opposite to the direction of 
if α and β have opposite signs.
This length equals either 
or
, depending on the directions of the vectors. If 
, then 
. The uniqueness on he multiplier l is evident: multiplying by different numbers, we obtain different vectors.
Definition.Suppose that 
and 
are vectors and ε is a point such that 
=
. Then the vector 
is called the sum of the vectors 
and 
and denoted by +
ε
 
A D 
C
Property 2.Addition of vectors is commutative; this means that, for any two vectors,
=
.
Property 3.Addition of vectors is associative; this means that, for any vectors,
.
Property 4.Addition is distributive with respect to multiplication by a number; i.e., for any vectors and 
and any number a,
.
To prove these properties, it is sufficient to refer to the triangle rule:
.
Property 5.For any numbers a and b and any vector 
,
.
The vectors on both sides are collinear.
Suppose that a and b are of the same sign. Then the vectors aand b have the same direction, and the length is equal to the sum of their lengths, i.e., 
.
a
a
But 
; thus, in this case, the vectors 
and a+b have the same length.
Their direction coincides with that of the vectorif the sign of a and b is positive, and is opposite to that of 
if the sign is negative.
Definition. Free vectors are vectors which can be translated, which means that they do not depend on the head but are determined by direction and length.
B 
B1 B
A A1 A
Consider vectors
, their sum 
is determined by one vector, whose head coincides with that of the first vector and the tail with that of the last vector.
,
.
Definition. The ort-vector of a vector is the vector of unit length whose direction coincides with that of .
is the ort-vector of .
Subtraction of vectors can be considered as the addition of two vectors, the second of which is taken with the sign –:
.
Theorem.If two vectors 
and 
can be represented as
, (1)
Then these vectors are collinear, and vice versa, if two vectors are collinear, then they relation (1) holds.
Proof. Suppose that (1) holds. If 
, then 
have the same direction, and consequently, lie on the same straight line or on parallel lines, i.e., are collinear.
Conversely, suppose that vectors are collinear: 
÷÷
. Then they can be placed on the same straight line, and the first vector is longer than the other second one by a factor of 
. Consequently, 
.
Â
À φ
φ l
À1 Â1
Definition.The projection of a vector onto an axis is defined as the length of the interval whose endpoints are the projections of the endpoints of the vector onto this axis which is taken with the sign + if the angle between the vector and the axis is acute and with this sign – if this angle is obtuse:
.
The projection of the sum of vectors is equal to the sum of the projections of the vectors:
.
l
1. Suppose that n vectors 
are given. These n vectors are said to be linearly dependent if one of them can be represented as a linear combination of the other vectors:
.
2. These n vectors are said to be linearly dependent if there exist numbers 
(
for some i) such that they can be represented as
=0.