Question

Consider the vectors v1, v2, v3 in R2. Vectors v1 and v2 are parallel.How many solutions x, y does the system xv1 + yv2 = v3 have? Argue geometrically:

Answer 1

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