Problem 1QC: Describe the length and direction of the vector 5v relative to v. Problem 2QC Problem 3QC Problem 4QC: Given the points P(2.3) and Q(4, 1), find the components of PQ. Problem 5QC: Find vectors of length 10 parallel to the unit vector u35,45. Problem 6QC: Verify that the vector 513,1213 has length 1. Problem 7QC: Solve 3u | 4v = 12w for u. Problem 1E: Interpret the following statement: Points have a location, but no size or direction; nonzero vectors... Problem 2E: What is a position vector? Problem 3E: Given a position vector v, why are there infinitely many vectors equal to v? Problem 4E: Use the points P(3.1) and Q(7.1) to find position vectors equal to PQ and QP. Problem 5E: If u = u1, u2 and v = v1, v2, how do you find u + v? Problem 6E: Find two unit vectors parallel to 2,3. Problem 7E: Is 1,1 a unit vector? Explain. Problem 8E: Evaluate 3,1+2,4 and illustrate the sum geometrically using the Parallelogram Rule. Problem 9E Problem 10E: Express the vector v = v1, v2 in terms of the unit vectors i and j. Problem 11E: How do you compute |PQ| from the coordinates of the points P and Q? Problem 12E: The velocity of a kayak on a lake is v=2,2,22. Find the speed and heading of the kayak. Assume the... Problem 13E: Vector operations Refer to the figure and carry out the following vector operations. 17. Scalar... Problem 14E: Vector operations Refer to the figure and carry out the following vector operations. 14. Scalar... Problem 15E: Vector operations Refer to the figure and carry out the following vector operations. 19. Scalar... Problem 16E: Vector operations Refer to the figure and carry out the following vector operations. 20. Scalar... Problem 17E Problem 18E: Vector operations Refer to the figure and carry out the following vector operations. 22. Vector... Problem 19E: Components and magnitudes Define the points O(0, 0), P(3, 2), Q(4, 2), and R(6, 1). For each vector,... Problem 20E Problem 21E: Components and equality Define the points P(3, 1), Q(1, 2), R(1, 2), S(3, 5), T(4, 2), and U(6, 4).... Problem 22E: Components and equality Define the points P(3, 1), Q(1, 2), R(1, 2), S(3, 5), T(4, 2), and U(6, 4).... Problem 23E: Components and equality Define the points P(3, 1), Q(1, 2), R(1, 2), S(3, 5), T(4, 2), and U(6, 4).... Problem 24E: Vector operations Let u = 4, 2, v = 4, 6, and w = 0, 8. Express the following vectors in the form a,... Problem 25E: Vector operations Let u = 4, 2, v = 4, 6, and w = 0, 8. Express the following vectors in the form a,... Problem 26E: Vector operations Let u = 4, 2, v = 4, 6, and w = 0, 8. Express the following vectors in the form a,... Problem 27E: Vector operations Let u = 4, 2, v = 4, 6, and w = 0, 8. Express the following vectors in the form a,... Problem 28E: Vector operations Let u = 3, 4, v = 1, 1, and w = 1, 0. 36. Find |u + v + w|. Problem 29E: Vector operations Let u = 3, 4, v = 1, 1, and w = 1, 0. 35. Find |2v|. Problem 30E Problem 31E: Vector operations Let u = 3, 4, v = 1, 1, and w = 1, 0. 41. Which has the greater magnitude, u v or... Problem 32E: Find a unit vector in the direction of v = 6,8. Problem 33E Problem 34E Problem 35E: Find the vector v of length 6 that has the same direction as the unit vector 1/2,3/2 Problem 36E: Find the vector v that has a magnitude of 10 and a direction opposite that of the unit vector... Problem 37E: Designer vectors Find the following vectors. 73. The vector in the direction of 5, 12 with length 3 Problem 38E Problem 39E: How do you find a vector of length 10 in the direction of v = 3, 2? Problem 40E: Let v = 8,15. a. Find a vector in the direction of v that is three times as long as v. b. Find a... Problem 41E Problem 42E Problem 43E: Unit vectors Define the points P(4, 1), Q(3, 4), and R(2, 6). 45. Find two unit vectors parallel to... Problem 44E Problem 45E Problem 46E Problem 47E: Unit vectors a. Find two unit vectors parallel to v = 6i 8j. b. Find b if v=13,b is a unit vector.... Problem 48E: Vectors from polar coordinates Suppose O is the origin and P has polar coordinates (r, ). Show that... Problem 49E: Vectors from polar coordinates Find the position vector op if o is the origin and P has polar... Problem 50E Problem 51E: Find the velocity v of an ocean freighter that is traveling 30 south of east at 30 km/hr. Problem 52E Problem 53E: Airplanes and crosswinds Assume each plane flies horizontally in a crosswind that blows... Problem 54E Problem 55E: Airplanes and crosswinds Assume each plane flies horizontally in a crosswind that blows... Problem 56E: A boat in a current The water in a river moves south at 10 mi/hr. A motorboat travels due east at a... Problem 57E: Another boat in a current The water in a river moves south at 5 km/hr. A motorboat travels due east... Problem 58E Problem 59E: Boat in a wind A sailboat floats in a current that flows due east at 1 m/s. Due to a wind, the boats... Problem 60E Problem 61E Problem 62E Problem 63E Problem 64E Problem 65E: Explain why or why not Determine whether the following statements are true and give an explanation... Problem 66E: Equal vectors For the points A(3, 4), B(6, 10), C(a + 2, b + 5), and D(b + 4, a 2), find the values... Problem 67E: Vector equations Use the properties of vectors to solve the following equations for the unknown... Problem 68E: Vector equations Use the properties of vectors to solve the following equations for the unknown... Problem 69E Problem 70E: Solving vector equations Solve the following pairs of equations for the vectors u and v. Assume i =... Problem 71E Problem 72E Problem 73E Problem 74E: Ant on a page An ant walks due east at a constant speed of 2 mi/hr on a sheet of paper that rests on... Problem 75E: Clock vectors Consider the 12 vectors that have their tails at the center of a (circular) clock and... Problem 76E: Three-way tug-of-war Three people located at A, B, and C pull on ropes tied to a ring. Find the... Problem 77E: Additional Exercises 8185. Vector properties Prove the following vector properties using components.... Problem 78E: Additional Exercises 8185. Vector properties Prove the following vector properties using components.... Problem 79E: Vector properties Prove the following vector properties using components. Then make a sketch to... Problem 80E: Vector properties Prove the following vector properties using components. Then make a sketch to... Problem 81E: Vector properties Prove the following vector properties using components. Then make a sketch to... Problem 82E Problem 83E: Magnitude of scalar multiple Prove that |cv| = |c| |v|, where c is a scalar and v is a vector. Problem 84E: Equality of vectors Assume PQ equals RS. Does it follow that PR is equal to QS? Explain your answer. Problem 85E: Linear independence A pair of nonzero vectors in the plane is linearly dependent if one vector is a... Problem 86E: Perpendicular vectors Show that two nonzero vectors u = u1, u2 and v = v1, v2 are perpendicular to... Problem 87E: Parallel and perpendicular vectors Let u = a, 5 and v = 2, 6. a. Find the value of a such that u is... Problem 88E: The Triangle Inequality Suppose u and v are vectors in the plane. a. Use the Triangle Rule for... format_list_bulleted