Find the orthogonal projection of upsilon 10 10 14 onto the

Find the orthogonal projection of upsilon = [-10 -10 14] onto the subspace W of R^3 spanned by [-4 -6 -2] AND [-6 -3 21]. proj_w(upsilon) = []

Solution

Subspace of W spanned by v1 and v2 :

v1.v1 = 16 +36 + 4 = 56 ; v1.v2 = 24 +18 -42 = 0 .They are orthogonal ; v2.v2 = 36 +9 + 441 = 486

u1 = v1/ |v1| = (-4/sqrt56 , -6/sqrt56, -2/sqrt56) ; u2 = v2/|v2| = ( -6/sqrt486 , -3/sqrt486 , 21/sqrt486 )

v.u1( dot product) = (-10, -10, 4).(-4/sqrt56 , -6/sqrt56, -2/sqrt56) ;

v.u2 ( dot product) = (-10, -10, 4).( -6/sqrt486 , -3/sqrt486 , 21/sqrt486 )

Projection of v can be found by a formula = (v.u1)u1 + (v.u2)u2