Trivial Questions Linear Algebra

Question

If $x$ and $y$ are perpendicular vectors and $\left\|x\right\|=2$ and $\left\|y\right\|=1$, why isn't $\left\|x+y\right\|$ simply $3$?

Would $\left\|x+y\right\|$ form the diagonal of the rectangle, and we use the Pythagorean theorem to solve?

Answer 1

As you said, use pythagoras theorem. $$\|x+y\|^2=\langle x+y, x+y\rangle=\|x\|^2+\|y\|^2+2\langle x, y \rangle =\|x\|^2+\|y\|^2$$

We shouldn't expect it to be equal to $3$.