1. Let the vectors, A, B, and C be defined by: A = 3i + 6j, B = i' + 43, and C = 5i 2j. (a) What is the magnitude of vector A? (b) Sketch the vector subtraction problem B C and find the components of the resultant vector. (c) Find the components of A + 2B. ((1) Express A B C as a magnitude and angle relative to the horizontal. 2. An ant walks 20 cm due north, 30 cm due east, and then 14 cm northeast. (a) Assuming that each portion of the ant's journey is a vector, sketch the ant's path. (b) How far has the ant travelled from its original position? (c) If the ant does not want to travel farther than 80 cm from its original position, how march farther north could it walk? 3. Consider an airplane taking off at an angle of 10 relative to the horizontal. After 2 s, the airplane has travelled a total of 140 m through the air. (a) Express the position, p, of the airplane compared with the point of liftoff as a vector using components and the i' andj basis vectors. (b) If the airplane had been taking off in a headwind, the position of the airplane would instead have been p' = p + w, in which the impact of the wind, w, is the vector w = 3oi'. Draw a sketch to Show the relationship between the vectors p', p, and w. (c) What angle would the plane make with the horizontal as a result of the headwind after 2 s? 4. As a boat travels in a river, its velocity, v, is determined by the current, c, and its relative velocity compared to the water, r. For instance, at one point in the boat's journey, these vectors are related in the following way: C (a) Write a vector equation that shows how v is related to c and r. (b) If the angle between v and c is 90, as shown, write an equation for the magnitude of v as a function of c and r. (c) At a different point in the boat's journey, if the current is owing due east at 5 m/s and the boat wants to travel due south at 10 m/s, what should the captain set its relative velocity to be (magnitude and direction)