1. Fill in the blanks. Consider the following vector v from (0, 0) to (a,b). v = ai + bj The horizontal component of v is _. The vertical component of v is _. The magnitude of v is given by The horizontal component of v is (1) The vertical component of v is (2) The magnitude of v is given by | |v | = (3) - (1) O b. (2) O a. (3) O vatb. a. Ob. O V(ai)2 + (bj)2 . ai. Obj. O Vaz + 62 . O bj. O ai. O vai + bj . 2. Complete the sentence below. If v= (a,b) is an algebraic vector whose initial point is the origin, then v is called a(n) vector. A vector whose initial point is the origin, then v is called a(n) (1) vector. (1) O force O unit O position O velocity nonzero3. Fill in the blanks. Vector v with initial point P, = (X1 .y1 ) and terminal point P2 = (X2,y2 ) is equal to the following vector. V= i+ Vector v with initial point P, = (x1 ,y, ) and terminal point P2 = (x2,y2 ) is equal to the following vector. V= ((1) ji + ((2) (1) O y2 - y1 (2) O X2 - X1 O y1 - 12 O X1 - X2 O X1 - X2 O y2 - y1 O X2 - X1 O y1 - y24. Fill in the blanks. Let v be a nonzero vector. If U is the direction angle measured from the positive xaxis to v, then the vector can be expressed in terms of its magnitude and direction angle as follows. V= ||\\-'||_|+ "VIl Let v be a nonzero vector. If H is the direction angle measured from the positive xaxis to v, then the vector can be expressed in terms of its magnitude and direction angle as follows. '4': "V\" [1) i + "VII (2} i (1} C:- cose (2) [:3 tanB C3 tanB r i slnB r:- sinB E_.- cost] 5. Complete the sentence below. If F1 and F2 are two forces simultaneously acting on an object. the vector sum F1 + F2 is called the force. If F1 and F2 are two forces simultaneously acting on an object. the vector sum F1 + F2 is called the (1) force. (1} -i_ _J effective E 3- resulting . J 5. ':- resultant If:- sum '. 6. Let v be the vector from initial point P1 = ( -2, - 9} to terminal point P2 = (7,4). Write v in terms ofi and j. The vector from point P1 = l - 2. - 9) to point P2 = (7.4) is v = :l. (Simplify your answer. Type your answer in terms of i and j.) 7. Let u = 9i - 5j and v = - 10i + 8j. Find the vector u + v. u+V= (Simplify your answer. Type your answer in terms of i and j.) 8. Let u =2i - 6j, and v = - 5i + 7j. Find u - v. U - VE (Type your answer in terms of i and j.) 9. Let v= - 6i + 5j, and w = - i-7j. Find 2w + 5v. 2w + 5v = (Type your answer in terms of i and j.) 10. Let v = - 8i + 6j and w = - i-6j. Find 2v - 7w. 2v - 7w = (Simplify your answer. Type your answer in terms of i and j.)11. Let u = 4i - 2j, and w = - i -6j. Find |w - ull. |/w - ull = (Type an exact answer, using radicals as needed.) 12. Find the unit vector that has the same direction as the vector v. v = 10i + 24j The unit vector that has the same direction as the vector v is (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression. Type your answer in terms of i and j. Rationalize all denominators.) 13. Find the unit vector that has the same direction as the vector v. v = 13i - 5j The unit vector that has the same direction as the vector v = 13i - 5j is (Simplify your answer, including any radicals. Type your answer in the form ai + bj. Use integers or fractions for any numbers in the expression. Rationalize all denominators.) 14. Write the vector v in terms of i and j whose magnitude | v| and direction angle 0 are given. Iv// = 6, 0 =60 V (Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression. Type your answer in the form ai + bj. Rationalize all denominators)15. Write the vector v in terms of i and j whose magnitude and direction angle are given. Ivil = =,0=113 (Type your answer in terms of i and j. Round to the nearest hundredth as needed.) 16. Let u = -4i + 5j, v =2i -j, and w = -7i. Find 4u - (2v - w). 4u - (2v - W) = (Type your answer in terms of i and j.) 17. Let u = -7i + 8j and v = 2i -j. Find |u + v||~ - |/u-v//2. lu + v/2 - 1/u-v/12 =1 18. Find the magnitude | v|| and the direction angle 0 for the given vector v. v= - 9i + 11j Ivll = (Round to the nearest hundredth as needed.) 0= (Round to the nearest tenth as needed.)19. The forces F1. F2, F3, Fn acting on an object are in equilibrium if the resultant force is the zero vector. F1+F2+F3+---+Fn= The forces F1 = 8i -8j and F2 = 6i +2j are acting on an object. 3. Find the resultant force. is. What additional force is required for the given forces to be in equilibrium? a. The resultant force is :|. {Simplify your answer. Type your answer in terms of i and j.) b. The additional force required for the given forces to be in equilibrium is :l. {Simplify your answer. Type your answer in terms of i and j.) 20. The figure shows a box being pulled up a ramp inclined at 12" from the horizontal. Use the following information to solve the problem. 120 120 BA = force of gravity, |BA = weight of the box AC = magnitude of the force needed to pull the box up the ramp BC = magnitude of the force of the box against the ramp If a force of 20 pounds is needed to pull the box up the ramp, find the weight of the box. The weight of the box is approximately pounds. (Type an integer or decimal rounded to the nearest tenth as needed.)