Multiplying vector with a number

author: Martin Marinov
viewed: 1326 times
Българска версия:

D: Product of the vector a^→ with the number x is the vector b^→ with the following properties:
1. |b^→|=|x|.|a^→|
2. b^→ and a^→ are in one direction if a^→ is not 0^→ and x > 0
    b^→ and a^→ are in different directions if a^→ is not 0^→ and x < 0

If x =0 or a^→ = 0^→ xa^→ = 0^→

Consequence 1: If AB^→ and CD^→ are non-zero vectors, then a non-zero number x exist so that AB^→ = xCD^→ if and only if AB^→ and CD^→ lie on one line or on parallel lines

Consequence 2: The points O, A and B lie on one line only when a number x exists and OA = xOB^→

Properties of the product of a vector with a number:
x.(a^→ +b)^→ =x.a^→ +x.b^→ , 1.a^→ =a^→ , (x+y)a^→ =x.a^→ +y.a^→ , x(y.a^→ )=(x.y.).a^→

Basic problem: If O is a scalene point, then M is on the middle of AB only when OM^→ =1/2(OA^→ +OB^→ )