Dot products As a rule, - will only be used to denote the dot product of vectors, and not vector scaling or multiplication of real numbers. Vector scaling will be denoted A5, where A (the Greek letter 'lambda') denote-1 a real number; for instance, 26 denote-1 scaling by 2, whereas 2 '1'! is meaningless. 1. Which of the following expressions is meaningful, and which is meaningless? All of if, 111', 21' are vectors in R3. a)('5' ('5' U" E: C -w+u. ) )17' d) lllli'i' 8) [17' MN 13)- 1?! (13 + 11'}. 2. Compute 1? - all for the following pairs of vectors. a) (2, 3) - (2, 3). l (2' 3) ' (3' 2)- ) (2. 3) - {~3,2). d) (2. 3) - (Ht)- 3. Determine whether the angle between the following vectors is acute, perpendicular. or obtuse. Do not attempt to compute the actual value of the angle. (In particular, do not use a calculator.) a) The vectors (2, 3) and (3, 2). b) The vectors (1, 4, 2) and (3,1,1). c) The vectors (1,1,1,1) and (1, 2,3, 4). d) The vectors (2, 1) and (2, 4). 4. Suppose you know that 13' and 15 meet at an angle of 17/3 radians. Further, suppose you know that \"ll - 3 and ||1I3|| .... 4. With no further information, compute the dot product 13' ' 1.3. 5. Find the value of m so that (1, 1:, 3) and (2, 1, 1) are perpendicular. 6. Find all vectors (1:,y, 2) which is perpendicular to both (1, U, 1) and (U, 1, 1). (There are many.) 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'\