When a vector is multiplied by a number l, each coordinate of this vector is multiplied by this number.

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 ОМ₁М₂ 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:

М₁(7;4;–3); М₂(1;–2;–2);

 

={–6; –6; 1}; ={6; 6; –1}.

 

Problem 2. Find the length of a vector :

.

From the right triangle ОМ₁М₂ , we find the hypotenuse

,

 

z

 

where .

M₁

From the other right triangle ОАМ₂ , 0 z₁ y

we find the hypotenuse .

Substituting it into А x₁

the first hypotenuse, we obtain x y₁ M₂

.

Thus, the length of a vector is defined by the formula

. (3)