Let u be the vector with initial point P3 1 and terminal poi

Let u be the vector with initial point P(3 -1) and terminal point Q(-3 p). (a) Graph u in the coordinate plane. (b) Express u in terms of i and j. (c) Find the length of u. Let u = (1, 3) and let v = (-6, 2). (a) Find u - 3v. (b) Find |u + v|. (c) Find u middot v. (d) Are u and v perpendicular? Let u = (-4 Squareroot 3 4). (a) Graph u in the coordinate plane, with initial point (0, 0). (b) Find the length and direction of u. A river is flowing due cast at 8 mi/h. A man heads his motorboat in the direction N 30 degree E in the river. The speed of the motorboat relative to the water is 12 mi/h. (a) Express the true velocity of the motorboat as a vector. (b) Find the true speed and direction of the motorboat. Let u = 3i + 2j and v = 5i - j. (a) Find the angle between u and v. (b) Find the component of u along v. (c) Find proj_v u. Find the work done by the force F = 3i - 5j in moving an object from the point (2.2) to the point (7, -13). Let P(4 3 -1) and Q(6, -1, 3) be two points in three-dimensional space. (a) Find the distance between P and Q. (b) Find an equation for the sphere whose center is P and for which the segment PQ vector is a radius of the sphere. (c) The vector u has initial point P and terminal point Q. Express u both in component form and using the vectors i j and k.

Solution

Solution:(4)

a)

Total velocity in North direction = 12*cos30° = 10.4 mi/h
Total velocity in East direction = 12*sin30° + 8 =14 mi/h

V² = 10.4² + 14²
V² = 108.16 + 196 = 304.16
Velocity V = Sqrt(304.16) = 17.44 mi/h

b)

Direction = tan-1 (10.4/14) = 36.6 E of N