Linear Operations on Vectors

Definition. The product of a vector and a real number lis the vector defined by the following conditions:

(1) ;

(2) The vector is collinear to ;

3. The vectors and have the same direction if λ >0and 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)

 

Definition. Suppose that and are vectors and E is a point such that = . Then the vector is called the sum of the vectors and and denoted by +

 

E ε

 

В

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,

.

Property 5. For any numbers a and b and any vector ,

.

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 –:

.

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:

.