A pair of nonzero vectors in the plane is linearly dependent if one vector is a scalar multiple of the other. Otherwise, the pair is linearly independent. 

a. Which pairs of the following vectors are linearly dependent and which are linearly independent: u = (2, -3), v = (-12, 18), and w = (4, 6)?

b. Geometrically, what does it mean for a pair of nonzero vectors in the plane to be linearly dependent? Linearly independent? 

c. Prove that if a pair of vectors u and v is linearly independent, then given any vector w, there are constants c₁ and c₂ such that w = c₁u + c₂v.