Homework SourseFind the orthogonal projection of v 8 11 15 onto the subsp
Find the orthogonal projection of v = [-8 - 11 -15] onto the subspace V of R^3 spanned by [6 4 -1] and [-2 -6 -36] proj_V(v) = ![Find the orthogonal projection of v = [-8 - 11 -15] onto the subspace V of R^3 spanned by [6 4 -1] and [-2 -6 -36] proj_V(v) = Solutionv1.v1 = 36 +24 +1 = 61 ;](/WebImages/43/find-the-orthogonal-projection-of-v-8-11-15-onto-the-subsp-1133664-1761606246-0.webp "Find the orthogonal projection of v = [-8 - 11 -15] onto the subspace V of R^3 spanned by [6 4 -1] and [-2 -6 -36] proj_V(v) = Solutionv1.v1 = 36 +24 +1 = 61 ;")
Solution
v1.v1 = 36 +24 +1 = 61 ; |v1| = sqrt61 = 7.81
v2.v2 = 4 +36 + 1296 = 1336 ; |v2| =sqrt1396 = 36.55
v1.v2 = -12 -24 +36 =0
they are orthognal (-8 , 11, -15)
unit vector u1 = v1/|v1| = ( 0.76 , 0.51 , -0.13)
u2 = v2/|v2| = ( -0.054 , -0.164 , -0.98 )
So, v.u1 = 1.48 ; v.u2 = 13.328
the projection of vcan be found by a formula
= 1.48u1 + 13.328u2
