Suppose v1v2v3 is an orthogonal set of vectors in R5 Let w b

Suppose v1,v2,v3 is an orthogonal set of vectors in R5.

Let w be a vector in Span(v1,v2,v3) such that v1v1=35,v2v2=66,v3v3=16, wv1=105,wv2=264,wv3=64,

then w= ___v1+ ___v2+ ___v3.

w = k1*v1 + k2*v2 + k3*v3. Use this in the second equation:

-105 = <w,v1> = <k1*v1+k2*v2+k3*v3,v1> = <k1*v1,v1> = k1*<v1,v1> = k1*35 =>

k1 = -105/35 =. -3

Similarly :
264 = <w,v2> = <k1*v1+k2*v2+k3*v3,v2> = <k1*v2,v2> = k2*<v2,v2> =k2*66

k2 = 264/66 = 4

64 = <w,v3> = <k1*v1+k2*v2+k3*v3,v3> = <k1*v3,v3> = k3*<v3,v3>

k3*16 = 64

k3 = 4

So, w = -3v1 +4v2 +4v3