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Transcript
Name: ! 5/29/15! ! ! ! ! ! ! ! ! ! ! AM(1)! ! PRACTICE FOR VECTORS AND PARAMETRICS TEST (12.1 - 12.3) ! Calculators OK. Give all answers to the nearest hundredth. USE OTHER PAPER!! ! 1. A fly begins at point A (5, –6) and crawls at a constant speed along a straight line on a coordinate plane. 3 seconds later the fly arrives at point B(–4, 9). Let t represent time in seconds. The velocity is in units/second.! ! a) Give vector and parametric equations for this line.! ! b) What is the fly’s speed?! ! c) When does the fly cross the x-axis?! ! d) What are the coordinates of the point at which the fly crosses the x-axis?! ! e) Write a cartesian equation for this situation.! ! ! 2. a) Sketch the curve defined by the parametric equations:! ! ! ! x = 5 cos (2t)! ! y = 5 sin (3t)! ! This describes the path of an object.! ! b) Label the starting position (t = 0) as point A.! ! c) At what time does the object pass through the point (–5, –5) ? ! ! d) What is the object’s position at time t = 225°? ! ! ! ! 3. A line is defined parametrically as follows:! x = 1 + 2t! y = 2 + 3t! For what value of t and at what point does the above line cross the line ! 3x – y = –5 ? Support your answer algebraically.! ! ! 4. A spider is crawling across a coordinate plane. The positive y-axis is oriented toward North. The spider begins at the point (5, 2) and crawls in a straight line at a speed of 10 units/minute on a bearing of 300°.! ! a) Write parametric equations describing the path of the spider. Let t represent time in minutes.! b) At what time and at what point does the spider cross the line x = –1 ?! ! 5. A spider is at (–6, 7) and is moving with a velocity of " 3,−2 . ! A beetle is at (–2, –5) and is moving with a velocity of " 2,1 .! a) The tracks of the spider and beetle cross at some point. What is that point?! b) Do the spider and the beetle meet in their travels? Justify your answer algebraically. Assume they meet if they come within 0.1 seconds of each other.! ! ! ! 6. An object travels at a constant speed along a straight line on a coordinate plane, beginning at the point (9, –8) and moving with a velocity of 5 units/sec. The object eventually passes through the point (–3, 8).! ! At what times and at what points does the object cross the circle whose equation is ( ) 2 " x − 3 + y2 = 25 (graphed below) ?! ! ! ! " ! 7. A spider is crawling across a coordinate plane. The positive y-axis is oriented toward North. The spider begins at the point (4, –1) and crawls in a straight line at a speed of 6 units/minute on a bearing of 40°.! ! ! ! ! ! ! ! A beetle begins at the point (7, 5) and moves along a straight line with a velocity of ! " 2,−6 (in units/minute).! Assuming they start moving at the same time, do the spider and the beetle meet in their travels ? (Assume that they meet if they pass the same point within 0.1 minutes of each other). Justify your answer algebraically.! ! ! ! ! ! ! 8. Given vectors in component form u = (-1, 3), v = (-2, 4), and w = (3, -2), simplify the following into a single vector in component form and find magnitude and direction . You may want to sketch the resultant to help you.! ! ! ! COMPONENT FORM ! ! MAGNITUDE ! ! DIRECTION! ! - v – w + 2u ! ! ! ____________! ! ____________! ! _________! ! 9. An airplane has an airspeed of 500 mph and is on a bearing of 260 . A wind is blowing at 35 mph due South. Find the resultant speed and bearing of the plane.! ! ! ! 10. A college student lobs a baseball out the 6th floor window of his dorm room. The position of the ball as a function of time is given by:! x = 20t y = 60 + 5t − 16t 2 ! ! When does the ball his the ground, and how far from the building does the ball hit the ground?! ! ! ! 11. A force F1 of 7 N is pulling an object in the compass direction of 150°. ! A force F2 of 10 N is pulling an object due west.! ! a) Make a vector diagram showing both force vectors and the resultant force vector F1 + F2.! b) Find the magnitude and direction of the resultant F1 + F2. Give answers to the nearest tenth.! ! c) Find the magnitude and direction of F3 such that F1 + F2 + F3 = 0 . ! ! ! ! 12. Ms. Pilch’s Hyundai is sitting on a steep hill, inclination " 13º . ! Her car weighs 2990 lbs.! ! a) Draw a sketch.! b) Find the force needed to keep the car from rolling down the hill.! c) Find the force perpendicular to the hill.
