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Mac 2313 Additional Problems Calculus III 1. An object in an xy-plane is being pulled by two ropes. One exerts a force in the direction of 2 ˆ i + 5 ˆ j and the other a force in the direction of 3 ˆ i + 2 ˆ j . The sum of the forces is 0 ˆ i + 190 ˆ j pounds. What are the magnitudes of the forces exerted by the ropes? Ans: 161.6 pounds and 72.1 pounds 2. Find a vector of length 8 in the direction of 2 i j +3 k , Ans: 16 14 r i 8 14 r j + 24 14 r k 3. Find two vectors of length 15 that are perpendicular to 5 ˆ i + 6 ˆ j in an xy-plane. Ans: ± 90 61 ˆ i + 75 61 ˆ j 4) Find the component of the vector 3i +5j in the direction of the vector from P to Q where the coordinates of P are (1,3)and Q are (5,1). Ans: .4i - .2j 5.The figure shows four forces, measured in newtons, that are applied to a ring. Find the magnitude and angle of inclination (the angle made with the positive x –axis) of their resultant. Ans: magnitude 2.98 with angle 7.99 degrees. 1.8 2.4 3 3.6 40 50 6. A 25 pound weight is suspended by 2 wires. One wire makes an angle of 45 ° with the vertical and the tension in it (the magnitude of the force it exerts) is 20 pounds. What is the tension in the other rope and what angle does it make with the vertical? Ans: Tension is 17.83 pounds and the angle is 0.92 rads.

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2 ˆ i 3 ˆ j + 4 ˆ k in the direction of the vector ˆ i + 2 ˆ j . Ans: -.8i + 1.6j 8) The force F = -100 j is exerted at a point P on an incline shown is the diagram below. Find the component of the F parallel to the incline. 3 5 x y answer: (-750/17)i + (-450,17) j 9) Find the angle between a diagonal through the center of a cube and one of its sides. answer : 54.7˚ 10. A cable is attached to the pole as shown below. The tension (force) in the cable is 500 lbs. A) Write a vector in component form for the tension at point P. B) What is the angle the cable makes with the ground in the direction from P to the base of the pole. 4 m
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