7. Suppose if and 15 are unit vectors which meet at an angle of ir/4 radians. Compute the length "I? + 13" of their sum. 8. Show that if 17+ iii and 6- ii are perpendicular, then "13'" = "13'". Challenge 1. The parallelogram law relates the lengths of the sides of a parallelogram to the diagonals. Prove the parallelogram law \"17+ 13"\" + "5- 1E"'ll2 = 2IIIill2 + 2ll'tll- Challenge 2. One of the most important and useful inequalities in math is the Cnnchy-Smrz inequality, which asserts |i - Iii] $ Il'llltiill (the dot product can't be bigger than the product of the lengths). 1. Prove the Cauchy-Schwarz inequality. 2. Use the Cauchy~Schwarz inequality to prove the triangle inequality "17+ ll $ llir' + EH. Gradient vectors 9. Find the gradient vector of n(z, y} - min! at the point (1,1). 10. Find the gradient vector of q(r, a) - rx/s + 1 at the point (1,0). Fbr the following two problems, nd the unit vector i? so that the directional derivative D5} is as large as posible at the given point. Then compute that directional derivative. 11. ung) = :csin(y) at (0, 1V3). 12. f(:l:, y, z) = 51:3 33:31 + Iyz at (3,4,5). Critical points and the second derivative test For #1317. nd the critical points of all of the following functions. Then apply the second derivative test to determine whether they are nondegenerate local maxim, uondegenerate local minima, saddle points, or degenerate critical points. 13- ay) = (I y)(1 9)- 14. ay) = I4 2::2 +3]:3 39. 15- at. 1;) = 9(8' - 1}- 16- aw) - (-'B2 + 9218"- 17. f(:r,y) u my +e'\