Problem 1E Problem 2E Problem 3E: Find the cross product a b and verify that it is orthogonal to both and a and b. 3.a = i + 3j 2k,... Problem 4E Problem 5E Problem 6E: Find the cross product a b and verify that it is orthogonal to both a and b. 6. a = ti + cos tj +... Problem 7E Problem 8E: If a = i 2k and b = j + k, find a b. Sketch a, b, and a b has vectors starting at the origin. Problem 9E: Find the vector, not with determinants, but by using properties of cross products. 9. (i j) k Problem 10E: Find the vector, not with determinants, but by using properties of cross products. 10. k (i 2j) Problem 11E: Find the vector, not with determinants, but by using properties of cross products. 11. (j k) (k ... Problem 12E: Find the vector, not with determinants, but by using properties of cross products. 12. (i + j) (i ... Problem 13E: State whether each expression is meaningful. If not, explain why. If so, state whether it is a... Problem 14E: Find |u v| and determine whether u v is directed into the page or out of the page. 14. Problem 15E: Find |u v| and determine whether u v is directed into the page or out of the page. 15. Problem 16E Problem 17E: If a = 2, 1, 3 and b = 4, 2, 1, find a b and b a. Problem 18E Problem 19E: Find two unit vectors orthogonal to both 3, 2, 1 and 1, 1, 0. Problem 20E: Find two unit vectors orthogonal to both j k and i + j. Problem 21E: Show that 0 a = 0 = a 0 for any vector a in V3. Problem 22E Problem 23E: Prove Property 1 of Theorem 8. 8 THEOREMIf a, b, and c are vectors and c is a scalar, then 1. a b =... Problem 24E: Prove Property 2 of Theorem 8. 8 THEOREMIf a, b, and c are vectors and c is a scalar, then 1. a b =... Problem 25E: Prove Property 3 of Theorem 8. 8 THEOREMIf a, b, and c are vectors and c is a scalar, then 1. a b =... Problem 26E: Prove Property 4 of Theorem 8. 8 THEOREMIf a, b, and c are vectors and c is a scalar, then 1. a b =... Problem 27E: Find the area of the parallelogram with vertices A(2, 1), B(0, 4), C(4, 2), and D(2, 1). Problem 28E: Find the area of the parallelogram with vertices K(1, 2, 3), L(1, 3, 6), M(3, 8, 6), and N(3, 7, 3). Problem 29E: (a) Find a nonzero vector orthogonal to the plane through the points P, Q, and R. and (b) find the... Problem 30E: (a) Find a nonzero vector orthogonal to the plane through the points P, Q, and R. and (b) find the... Problem 31E: (a) Find a nonzero vector orthogonal to the plane through the points P, Q, and R. and (b) find the... Problem 32E: (a) Find a nonzero vector orthogonal to the plane through the points P, Q, and R, and (b) find the... Problem 33E: Find the volume of the parallelepiped determined by the vectors a, b. and c. 33. a = 1, 2, 3), b =... Problem 34E: Find the volume of the parallelepiped determined by the vectors a, b. and c. 34. a = i + j , b = j +... Problem 35E: Find the volume of the parallelepiped with adjacent edges PQ, PR, and PS. 35. P(2, 1, 0), Q(2. 3.... Problem 36E: Find the volume of the parallelepiped with adjacent edges PQ, PR. and PS. 36. P(3, 0, 1), Q(1, 2,... Problem 37E: Use the scalar triple product to verify that the vectors u = i + 5j 2k, v = 3i j, and w = 5i + 9j ... Problem 38E Problem 39E Problem 40E Problem 41E Problem 42E: Let v = 5j and let u be a vector with length 3 that starts at the origin and rotates in the... Problem 43E: If a b = 3 and a b = 1, 2, 2, find the angle between a and b. Problem 44E: (a) Find all vectors v such that 1, 2, 1 x v = 3, 1, 5) (b) Explain why there is no vector v such... Problem 45E: (a) Let P be a point not on the line L that passes through the points Q and R. Show that the... Problem 46E: (a) Let P be a point not on the plane that passes through the points Q, R, and S Show that the... Problem 47E: Show that | a b |2 = | a |2| b |2 (a b)2 Problem 48E Problem 49E: Prove that (a b) (a + b) = 2(a b). Problem 50E: Prove Property 6 of Theorem 8, that is, a(bc)=(ac)b(ab)c 8 THEOREM If a, b, and c are vectors and c... Problem 51E: Use Exercise 50 to prove that a(bc)+b(ca)+c(ab)=0 50. Prove Property 6 of Theorem 8, that is,... Problem 52E: Prove that (ab)(cd)=|acbcadbd| Problem 53E: Suppose that a 0. (a) If a b = a c, does it follow that b = c? (b) If a b = a c, does it follow... Problem 54E: If v1, v2, and v3 are noncoplanar vectors, let k1=v2v3v1(v2v3) k2=v3v1v1(v2v3) k3=v1v2v1(v2v3)... format_list_bulleted