Decomposition of vectors.
Theorem 1. An arbitrary vector
Theorem 2. An arbitrary vector
Let The Cartesian system of coordinates. Consider the following coordinate system: take mutually perpendicular unit vectors
Definition. The triple of vectors From the triangle ОММ1, we obtain
Since the vector
From the triangle ОАМ1 , we obtain
because, by analogy, the vectors
Thus, the radius vector Consider the vectors
and their sum
Under addition the respective coordinates are added Let us multiply the vector
When a vector is multiplied by a number l, each coordinate of this vector is multiplied by this number. Example. Find the vector
Let us find the required vector in vector notation:
To find the same vector in vector notation, we multiply the first vector by 4 and the second by –3 and sum their coordinates:
Given two points Thus, we have found the required vector in the coordinate notation:
To find the coordinates of a vector, we must subtract the coordinates of its tail from the coordinates of the head. For example, let us find vectors with given coordinates of heads and tails: М 1(7;4;–3); М 2(1;–2;–2);
Find the length of a vector
From the right triangle ОМ 1 М 2, we find the hypotenuse
where M 1 From the other right triangle ОАМ 2, 0 z1 y we find the hypotenuse Substituting it into А x 1 the first hypotenuse, we obtain x y 1 M 2
Thus, the length of a vector is defined by the formula
|
in the plane can be decomposed into two noncollinear vectors:
.
in space can be decomposed into three noncoplanar vectors
.
be noncoplanar vectors.
and 
, draw coordinate axes x,y and z along them, and fix a unit on metric scale:
,
.
is called right if, looking from the endpoint of the last vector, we see that the shorter rotation from the first vector to the second is anticlockwise.
,
is collinear to the unit vector 
, it follows that
.
,
and 
are collinear to the unit vectors 
and 
. Substituting the vector 
thus obtained, we see that
. (2)
is represented as the sum of 
and multiplied by the corresponding coordinates of the point М.
and 
.
by a number l:
.
if
; 
.
.
.
and 
in space, find the vector 
.
. (2¢¢)
={–6; –6; 1}; 
={6; 6; –1}.
:
.
,
z
.
.
.
. (3)
