Fill in the Outline for the proof that the two definitions of the dot product are equivalent for two dimensional vectors.

• Let v₁ = <x₁, y₁> and v₂ = <x₂, y₂> be 2 arbitrary vectors. Place their tails together on the origin and call the angle between the vectors, θ.
• Draw a line segment that connects the heads of the vectors and call the length of the line segment, c.
  
  
  
  
  
  
• Use the Law of Cosines to finish the equation:
  (1)     c² = |v₁|² + _________________________
• Use the distance formula to write the right side of the equations:
  
  (2)     |v₁|² = _______________________
  
  (3)     |v₂|² = _______________________
  
  (4)     c² = ______________________
  
• Substitute the right side of equations (2), (3), and (4) into equation (1) and simplify.
  
  (5)     ____________________________________________________________
• Recall the definition of the dot product:
  
  (6)     v₁ • v₂ = _____________________________
• Substitute (6) into (5):
  
  ____________________________________________________________________
  

  ⌊QED