We want to see how many solutions has an equation given some restrictions on the vectors of the equation.
We have 3 vectors in R2.
v₁, v₂, and v₃.
Where we know that v₁ and v₂ are parallel. And two vectors are parallel if one is a scalar times the other.
Then we can write:
v₂ = c*v₁
Where c is a real number.
Then our system:
x*v₁ + y*v₂ = v₃
Can be rewriten to:
x*v₁ + y*c*v₁ = v₃
(x + y*c)*v₁ = v₃
Assuming x, y, and c are real numbers, this only has a solution if v₁ is also parallel to v₃, because as you can see, the equation says that v₃ is a scallar times v₁.
Geometrically, this means that if we sum two parallel vectors, we will get a vector that is parallel to the two that we added.
If you want to learn more, you can read:
brainly.com/question/13322477
