Question

Suppose that u = < u1,u2 > and v = < v1,v2 > are vectors such that | u+v |^2 = | u |^2 + | v |^2. Prove that u and v are orthogonal.

Answer 1

If u= (u1,u2,u3) andv= (v1,v2,v3), then the dot product of u and v is u·v=u1v1+u2v2+u3v3. For instance, the dot product of u=i−2j−3kandv= 2j−kisu·v= 1·0 + (−2)·2 + (−3)(−1) =−1.

Properties of the Dot Product.

Let u,v, and w be three vectors and let c be a real number. Then u·v=v·u,(u+v)·w=u·w+v·w,(cu)·v=c(u·v).

Further, u·u=|u|2.  

Thus, if u=0is the zerovector, then u·u= 0, and otherwise u·u>0.1

Orthogonality Two vectors u and v are said to be  orthogonal(perpendicular), if the angle between them is 90◦.Theorem. Two vectors u and v are orthogonal if and only if u·v= 0.