Verified answer

First we want the projection p of w onto the subspace. Then orthog comp = w−p.

One way to get p is to derive an orthogonal basis for V. Then the projection of w is the sum of the projections of w onto each orthogonal basis vector.

To find the basis we choose the first vector as v₁=(2,1,2) and then use Gram-Schmidt

v₂ = (1,−1,1) – {v₁•(1,−1,1)}v₁/||v₁||² = (1,−1,1) − 3(2,1,2)/9 = ⅓(1,−4,1)

We can drop the ⅓ and take orthog basis for V as (2,1,2) & (1,−4,1)

∴ p = {(3,5,−1)•v₁}v₁/||v₁||² + {(3,5,−1)•v₂}v₂/||v₂||² = {9}v₁/9 + {−18}v₂/18

= (2,1,2) – (1,−4,1) = (1,5,1)

orthog component of w = (3,5,−1)−(1,5,1) = (2,0,−2)

Are these meant to be vectors ?