x. y - Fix n E N. Given x, y E R", recall that the dot product x y ER is defined by . Prove the following for any x, y, z = R" and a R: 1 (a) x. x = |x|| (b) x y =y.x (c) x (y +z) = x.y +x.z (d) (ax) y = a(x y) = x. (ay) (e) ||ax|| = |a| ||x|| (f) ||x y|| = ||x|| 2x y + ||y|| (g) If x = ||x|| . x, then |||| 1. = (h) x y ||x|| ||y|| (Cauchy-Schwarz in- . equality") (i) x +y|| Hint: For (h), first show that the statement is equivalent to x ||x || 0. For (i), use that squaring both sides yields an equivalent statement. x|| + ||y|| ||*|| ||||. Then use that