||x + y|| = 4 and ||x − y|| = 2. What is x · y? (Linear Algebra)

Question

• Background Information:

I am studying linear algebra, and for this question, I understand how to work with the formula of length/norm and dot product. However, I don't know how to approach this problem. I appreciate any hint, example, or explanation that can guide me to solve it. Thank you :)

• Question:

||x + y|| = 4 and ||x − y|| = 2. What is x · y?

Answer 1

$$\Vert x + y \Vert^2 = (x+y).(x+y) = x.x + 2x.y + y.y = 16 $$ $$\Vert x - y \Vert^2 = (x-y).(x-y) = x.x - 2x.y + y.y = 4 $$ Subtract second from first: $$2x.y + 2x.y = 16-4 =12$$ so $$x.y = \frac{12}{4} = 3$$

Answer 2

Guide:

$$\|x+y\|=4$$

If you square it,

$$\langle x+y, x+y\rangle =16$$

$$\|x\|^2+2\langle x, y\rangle +\|y\|^2=16$$

Do similar thing for the other equation as well and you caan solve for $\langle x, y \rangle$.