How to create vectors?

Solve the following using Cauchy Schwartz inequality. Remember to create two vectors.... Given x,y are Real numbers. If  2x +3y =4 find the value of  x and  y such that x^2 + y^2  has the minimum value?

• If anyone can give me some insight to this it would be much appreciated bc I have no idea what i'm doing...

• Answer:

  Consider the two vectors v1=⟨x,y⟩ v_1 = \langle x,y\rangle v2=⟨2,3⟩ v_2 = \langle 2,3\rangle Cauchy-Schwarz says that |v1⋅v2|≤∥v1∥⋅∥v2∥, |v_1\cdot v_2 | \leq \| v_1 \| \cdot \| v_2\|, which expanded out gives |2x+3y|≤x2+y2−−−−−−√⋅13−−√. |2x + 3y| \leq \sqrt{x^2+y^2} \cdot \sqrt{13}. Given that 2x+3y=42x+3y =4, we find x2+y2≥1613. x^2+y^2 \geq \frac{16}{13}. Furthermore, the Cauchy-Schwarz inequality also says that this inequality is actually an equality if and only if the vector v1v_1 is parallel to the vector v2 v_2; that is, there is some number zz such that x=2z x = 2z y=3z y = 3z But then since 2x+3y=42x+3y = 4 we find 13z=4,13z = 4, so z=413 z = \frac{4}{13}. We conclude x=813 x = \frac{8}{13} y=1213, y = \frac{12}{13}, and this is the unique minimizing solution.