If $\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}, \overrightarrow{\mathbf{c}}$ are three vectors…
If $\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}, \overrightarrow{\mathbf{c}}$ are three vectors such that $\overrightarrow{\mathbf{a}}=\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}$ and the angle between $\overrightarrow{\mathbf{b}}$ and $\overrightarrow{\mathbf{c}}$ is $\frac{\pi}{2}$, then:
$a^2=b^2+c^2$
$b^2=c^2+a^2$
$c^2=a^2+b^2$
$2 a^2-b^2=c^2$
Solution
Given that $\overrightarrow{\mathbf{a}}=\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}$
and $\overrightarrow{\mathbf{b}} \perp \overrightarrow{\mathbf{c}}$
then $(\overrightarrow{\mathbf{a}})^2=(\overrightarrow{\mathbf{b}})^2+(\overrightarrow{\mathbf{c}})^2$
$a^2=b^2+c^2$
