Please explain thank you Given v 4 8 4 find the coordinates

Please explain, thank you!

Given v = [-4 -8 4], find the coordinates for v in the subspace spanned by u_1 =[5 4 -1] and u_2 = [-4 -1 -24] Note that u_1 and u_2 are orthogonal.

Solution

Let,

v=au1+bu2

u1,u2 are orthogonal hence, u1.u2 =0

v.u=au1.u1

-20-32-4=a(5^2+4^2+1)

-56 =42 a

a=-4/3

v.u2=b u2.u2

(16+8-96)=b (16+1+576)

-72=593 b

b=-72/593