Vector Calculus, Fourth Edition, makes use of the language and notation of vectors and matrices to educate multivariable calculus. it's perfect for college kids with an exceptional history in single-variable calculus who're able to considering in additional basic phrases concerning the themes within the direction. this article is exceptional from others through its readable narrative, a number of figures, thoughtfully chosen examples, and thoroughly crafted workout units. Colley comprises not just easy and complicated workouts, but additionally mid-level workouts that shape an important bridge among the 2.
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23. Give a symmetric form for the line having parametric equations x = t + 7, y = 3t − 9, z = 6 − 8t. 24. A certain line in R3 has symmetric form x −2 y−3 z+1 = = . 5 −2 4 Write a set of parametric equations for this line. 25. Give a set of parametric equations for the line with symmetric form x +5 y−1 z + 10 = = . 3 7 −2 26. Are the two lines with symmetric forms x −1 y+2 z+1 = = 5 −3 4 and x −4 y−1 z+5 = = 10 −5 8 the same? Why or why not? 27. Show that the two sets of equations x −2 y−1 z x +1 y+6 z+5 = = and = = 3 7 5 −6 −14 −10 actually represent the same line in R3 .
20. Find three nonparallel vectors that are perpendicular to i − j + k. 21. Is it ever the case that proja b = projb a? If so, under what conditions? 22. Prove properties 2, 3, and 4 of dot products. 23. 4. 24. Suppose that a force F = i − 2j is acting on an object moving parallel to the vector a = 4i + j. Decompose F into a sum of vectors F1 and F2 , where F1 points along the direction of motion and F2 is perpendicular to the direction of motion. ) 26. A refrigerator is dragged 12 ft across a smooth floor using a rope and 60 lb of force directed along the rope.
Why or why not? 31. Do the parametric equations x = 5t − 1, y = 2t + 2 2 3, z = 1 − t 2 determine a line? Explain. 32. A bird is flying along the straight-line path x = 2t + 7, y = t − 2, z = 1 − 3t, where t is measured in minutes. (a) Where is the bird initially (at t = 0)? Where is the bird 3 minutes later? (b) Give a vector that is parallel to the bird’s path. (c) When does the bird reach the point 34 , 1 , − 11 ? 3 6 2 (d) Does the bird reach (17, 4, −14)? 17 Exercises 44. (a) Find the distance from the point (−2, 1, 5) to any point on the line x = 3t − 5, y = 1 − t, z = 4t + 7.
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