Vector Pythagorean Theorem
Recall that for a right triangle with side lengths $a$, $b$, and $c$ where $c$ is the hypotenuse, then $a^2 + b^2 = c^2$. We will prove a similar result with regards to vectors. Suppose that we have two vectors $\vec{u}$ and $\vec{v}$ that are perpendicular to each other, that is $\vec{u} \perp \vec{v}$. The vector $\vec{u} + \vec{v}$ will be the hypotenuse of our triangle:
 Theorem 1 (The Pythagorean Theorem for Vectors): If $\vec{u}, \vec{v} \in \mathbb{R}^n$ then $\| \vec{u} + \vec{v} \| ^2 = \| \vec{u} \| ^2 + \| \vec{v} \| ^2$.
• Proof: The prove is done algebraically while noting that $\vec{u} \cdot \vec{v} = 0$ since $\vec{u} \perp \vec{v}$.