## Elementary Linear Algebra 7th Edition

Given the vectors ${u}=(4,1,-5), \quad {v}=(2,-3,1)$, then their lengths can be calculated as follows $$\|{u}\|^2=16+1+25=42, \quad \|{v}\|^2=4+9+1=14.$$ Since $u+v=(6,-2,-4)$, then $$\|{u+v}\|^2=36+4+16=56.$$ One can see that, $$\|{u}+{v}\|^{2}=\|{u}\|^{2}+\|{v}\|^{2}$$ Hence, Pythagorean Theorem is verified.