Question

D1. If a and b are nonzero, then an orthonormal basis for the plane z = ax + by is

Answer 1

Here basically we use the Gram Schmidt process for orthogonalization for the basis vector of the given plane.

kant to find plane an orthonormal basis for the an+by, ato, 670. Firet we try to find that a basis for the given plane. we As know any plane is the set of vectore (miy,z), Retietying home properties. Here the plane Z= antby : = {(0.4.2) zzant by, 440, 6403 (m.y. 2) = (wy, antby) = (1,0, an) Hoy, by) n(1,0,0) + g (0,1,6) Thus. Thus the given plane is apan { (10,0). Colib)}

Obeerve that these two vectore are linearly independent. B = Now Thus the basis for the plane is {(1, 0, 4), (o, Đ] : 449,540 will perform Gram - Schmidt process to get the onthogonal basis 4,= (1,0,0), 4q = Now suppore (1.0,a). Uring the Gram- echmidt process: (uz" v) (0.1, b). Let V, = 4, = V2= 42- vi (wit) (ob) - [60,463" (10,41] - ((1,0,437(1,0,1)) (110,9) ab (1,0,4) = (016) - 1792 - Ginton =(0,1b) / ab 0 926 1492 Itazi

= ab il, b-026 1 +92 1+02 6 ab 1+92! 1, hoit we have the orthogonal basis Thus {w.v}={(1016). (Hina 1, b 192 Now the orthonormal basi N'= vi (1,0,0) ال۱۱۷ Nitar 10, Vita? Jitar and Na'= 11 Vall 1+92 1+07 ab tit 1 +92 1 +92 (1+2) a? 6? sta? +62 + (1+1) ab b Ha² tal

(+62) a²6²+6²+3a²+94+1 [E ab Ha? Ha? ab ELE Ita² q?6²+64 39²7 94+1 - Q²6² +6² +39²+ a4+1 b a²6² +6² +69² +94 + Thus, Let m= 5 926'+b? +39?+Q4+1 ab 1 +92 then vis) Thus ab 1+a? a 6, - frirvi} = {(fita Sitat یا y the orithonormal besie for the plane