Remark.Relation (2) is a written in vector notation; the coordinate notation of a vector is
. (2¢)
Example. Find the vectorif
; .
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:
.
Problem 1.Given two points and in space, find the vector . Consider the radius-vectors
z M1M2 ,
.
y
x
It is seen from the triangle ÎÌ1Ì2 that , where
Thus, we have found the required vector in the coordinate notation:
. (2¢¢)
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:
Ì1(7;4;–3); Ì2(1;–2;–2);
={–6; –6; 1}; ={6; 6; –1}.
Problem 2. Find the length of a vector :
.
From the right triangle ÎÌ1Ì2 , we find the hypotenuse
,
z
where .
M1
From the other right triangle ÎÀÌ2 , 0 z1 y
we find the hypotenuse .
Substituting it into À x1
the first hypotenuse, we obtain x y1 M2
.
Thus, the length of a vector is defined by the formula
. (3)