Let a, b be vectors. Show that $a \cdot b = \frac{1}{4} (|a+b|^2 − |a−b|^2)$

Question

Let $a, b$ be vectors. Show that $$a \cdot b = \frac{1}{4} (|a+b|^2 − |a−b|^2)$$

I am trying to write this proof for vectors and I am unsure how to start.

I know the magnitude of a vector $U$ is the square root of the sum of the elements squared or $||U|| = \sqrt{U \cdot U}$

However, I do not know any mathematical rules for $|a−b|$ as compared to $|a+b|$ and the significance of these magnitudes squared. I am definitely missing something.

I appreciate any and all help.

Answer 1

We have $$|\vec{a}+\vec{b}|^2=|\vec{a}|^2+|\vec{b}|^2+2\vec{a}\cdot\vec{b}$$