If v1,v2,v3v_1, v_2, v_3v1,v2,v3 are orthogonal, what is ∥v1+v2+v3∥2\|v_1 + v_2 + v_3\|^2∥v1+v2+v3∥2?
∥v1∥2+∥v2∥2+∥v3∥2\|v_1\|^2 + \|v_2\|^2 + \|v_3\|^2∥v1∥2+∥v2∥2+∥v3∥2
(∥v1∥+∥v2∥+∥v3∥)2(\|v_1\| + \|v_2\| + \|v_3\|)^2(∥v1∥+∥v2∥+∥v3∥)2
∥v1∥2+∥v2∥2+∥v3∥2+2(v1⋅v2+v2⋅v3+v1⋅v3)\|v_1\|^2 + \|v_2\|^2 + \|v_3\|^2 + 2(v_1 \cdot v_2 + v_2 \cdot v_3 + v_1 \cdot v_3)∥v1∥2+∥v2∥2+∥v3∥2+2(v1⋅v2+v2⋅v3+v1⋅v3)
0