Use the inner product ⟨f,g⟩=f(−2)g(−2)+f(0)g(0)+f(1)g(1) in P2 to find the orthogonal projection of f(x)=2x2+3x+8 onto the line L spanned by g(x)=3x2−5x−9.

Answer 1

Solution,

By definition, proj _L f = <f, g> ⁄ |g|² · g

Now, f(x) = 2x² + 3x +8 and g(x) = 3x² - 5x - 9

<f, g> = f(-2)g(-2) + f(0)g(0) + f(1)g(1)

= 10(13) + 8(-9) + 13(-11)

= -85

|g|² = g(-2)² + g(0)² + g(-1)²

= 13² + (-9)² + (-11)²

= 371

∴ proj _L f = (-85) ⁄ 371 ⋅ (3x² - 5x - 9)

= -255 ⁄ 371 x² + 425 ⁄ 371 x + 765 ⁄ 371 ← Answer