Question

How is the norm of a vector related to the Theorem of Pythagoras?

Answer 1

We denote the length of a vector $\mathbf{v}$ by the symbol $\|\mathbf{v}\|$, which is read as the norm of $\mathbf{v}$, the length of $\mathbf{v}$, or the magnitude of $\mathbf{v}$ (the term "norm" being a common mathematical synonym for length). It follows from the Theorem of Pythagoras that the norm of a vector $\left(v_{1}, v_{2}\right)$ in $R^{2}$ is $$ \|\mathbf{v}\|=\sqrt{v_{1}^{2}+v_{2}^{2}} $$ Similarly, for a vector $\left(v_{1}, v_{2}, v_{3}\right)$ in $R^{3}$, it follows and two applications of the Theorem of Pythagoras that $$ \|\mathbf{v}\|^{2}=(O R)^{2}+(R P)^{2}=(O Q)^{2}+(Q R)^{2}+(R P)^{2}=v_{1}^{2}+v_{2}^{2}+v_{3}^{2} $$ and hence that $$ \|\mathbf{v}\|=\sqrt{v_{1}^{2}+v_{2}^{2}+v_{3}^{2}} $$