Problem 1E: Express the dot product of u and v in terms of their magnitudes and the angle between them. Problem 2E: Express the dot product of u and v in terms of the components of the vectors. Problem 3E: Compute 2, 3, 6 1, 8, 3. Problem 4E Problem 5E Problem 6E Problem 7E Problem 8E Problem 9E Problem 10E Problem 11E Problem 12E Problem 13E Problem 14E Problem 15E Problem 16E Problem 17E Problem 18E Problem 19E Problem 20E Problem 21E Problem 22E Problem 23E Problem 24E Problem 25E: Sketching orthogonal projections Find projvu and scalvu by inspection without using formulas. 25. Problem 26E: Sketching orthogonal projections Find projvu and scalvu by inspection without using formulas. 26. Problem 27E: Sketching orthogonal projections Find projvu and scalvu by inspection without using formulas. 27. Problem 28E: Sketching orthogonal projections Find projvu and scalvu by inspection without using formulas. 28. Problem 29E: Calculating orthogonal projections For the given vectors u and v, calculate projvu and scalvu. 29. u... Problem 30E: Calculating orthogonal projections For the given vectors u and v, calculate projvu and scalvu. 30. u... Problem 31E Problem 32E Problem 33E: Calculating orthogonal projections For the given vectors u and v, calculate projvu and scalvu. 33. u... Problem 34E: Calculating orthogonal projections For the given vectors u and v, calculate projvu and scalvu. 34. u... Problem 35E Problem 36E: Calculating orthogonal projections For the given vectors u and v, calculate projvu and scalvu. 36. u... Problem 37E Problem 38E: Computing work Calculate the work done in the following situations. 38. A stroller is pushed 20 m... Problem 39E Problem 40E: Computing work Calculate the work done in the following situations. 40. A constant force F = 4, 3, 2... Problem 41E: Computing work Calculate the work done in the following situations. 41. A constant force F = 40, 30... Problem 42E Problem 43E Problem 44E: Parallel and normal forces Find the components of the vertical force F = 0, 10 in the directions... Problem 45E: Parallel and normal forces Find the components of the vertical force F = 0, 10 in the directions... Problem 46E: Parallel and normal forces Find the components of the vertical force F = 0, 10 in the directions... Problem 47E Problem 48E Problem 49E: Orthogonal vectors Let a and b be real numbers. 49. Find all vectors 1, a, b orthogonal to 4, 8, 2. Problem 50E Problem 51E Problem 52E: Orthogonal vectors Let a and b be real numbers. 52. Find two vectors that are orthogonal to 0, 1, 1... Problem 53E Problem 54E: Vectors with equal projections Given a fixed vector v, there is an infinite set of vectors u with... Problem 55E: Vectors with equal projections Given a fixed vector v, there is an infinite set of vectors u with... Problem 56E: Vectors with equal projections Given a fixed vector v, there is an infinite set of vectors u with... Problem 57E: Vectors with equal projections Given a fixed vector v, there is an infinite set of vectors u with... Problem 58E: Decomposing vectors For the following vectors u and v, express u as the sum u = p + n, where p is... Problem 59E: Decomposing vectors For the following vectors u and v, express u as the sum u = p + n, where p is... Problem 60E: Decomposing vectors For the following vectors u and v, express u as the sum u = p + n, where p is... Problem 61E: Decomposing vectors For the following vectors u and v, express u as the sum u = p + n, where p is... Problem 62E Problem 63E Problem 64E Problem 65E Problem 66E: Orthogonal unit vectors in 3 Consider the vectors I=1/2,1/2and J=1/2,1/2. 66. Show that I and J are... Problem 67E: Orthogonal unit vectors in 3 Consider the vectors I=1/2,1/2and J=1/2,1/2. 67. Express I and J in... Problem 68E: Orthogonal unit vectors in 3 Consider the vectors I=1/2,1/2and J=1/2,1/2. 68. Write the vector 2, 6... Problem 69E: Orthogonal unit vectors in 3 Consider the vectors I=1/2,1/2,1/2, J=1/2,1/2,0, and K=1/2,1/2,1/2. a.... Problem 70E: Angles of a triangle For the given points P, Q, and R, find the approximate measurements of the... Problem 71E: Angles of a triangle For the given points P, Q, and R, find the approximate measurements of the... Problem 72E: Flow through a circle Suppose water flows in a thin sheet over the xy-plane with a uniform velocity... Problem 73E: Heat flux Let D be a solid heat-conducting cube formed by the planes x = 0, x = 1, y = 0, y = 1, z =... Problem 74E: Hexagonal circle packing The German mathematician Gauss proved that the densest way to pack circles... Problem 75E: Hexagonal sphere packing Imagine three unit spheres (radius equal to 1) with centers at O(0, 0, 0),... Problem 76E: Properties of dot products Let u = u1, u2, u3, v = v1, v2, v3, and w = w1, w2, w3. Prove the... Problem 77E Problem 78E Problem 79E Problem 80E: Properties of dot products Let u = u1, u2, u3, v = v1, v2, v3, and w = w1, w2, w3. Prove the... Problem 81E Problem 82E Problem 83E: Direction angles and cosines Let v = a, b, c and let , , and be the angles between v and the... Problem 84E Problem 85E Problem 86E: CauchySchwarz Inequality The definition u v = |u| |v| cos implies that |u v| | u| |v| (because |... Problem 87E: CauchySchwarz Inequality The definition u v = |u| |v| cos implies that |u v| | u| |v| (because |... Problem 88E: CauchySchwarz Inequality The definition u v = |u| |v| cos implies that |u v| | u| |v| (because |... Problem 89E: Diagonals of a parallelogram Consider the parallelogram with adjacent sides u and v. a. Show that... Problem 90E format_list_bulleted