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2 Dot Product 15 18. Use the Cauchy-Schwarz inequality to solve the following max/min problem: If the (long) diagonal of a rectangular box has length c, what is the greatest the sum of the length, width, and height of the box can be? For what shape box does the maximum occur? 19. Give an alternative proof of the Cauchy-Schwarz inequality, as follows. Let a = |x||, b = ||y||, and deduce from ||bx - ay l|20 that x y ab. Now how do you show that |x yl ab? When does equality hold? 20. (a) Letx and y be vectors with ||x || = lly|l. Prove that the vector x + y bisects the angle between x and y ||x || and b = ||y||. Prove that (b) More generally, if x and y are arbitrary nonzero vectors, let a the vector bx + ay bisects the angle between x and y. 21. Use vector methods to prove that the diagonals of a parallelogram bisect the vertex angles if and only if the parallelogram is a rhombus. 22. Given AABC with D on BC as shown in Figure 2.6. Prove that if AD bisects ZBAC, then BD|/CD| ||AB ||/|| A || . (Hint: Use Exercise 20b. Let x = AB and y = AC; give two expressions for AD in terms of x and y and use Exercise 1.1.10.)

2 Dot Product 15 18. Use the Cauchy-Schwarz inequality to solve the following max/min problem: If the (long) diagonal of a rectangular box has length c, what is the greatest the sum of the length, width, and height of the box can be? For what shape box does the maximum occur? 19. Give an alternative proof of the Cauchy-Schwarz inequality, as follows. Let a = |x||, b = ||y||, and deduce from ||bx - ay l|20 that x y ab. Now how do you show that |x yl ab? When does equality hold? 20. (a) Letx and y be vectors with ||x || = lly|l. Prove that the vector x + y bisects the angle between x and y ||x || and b = ||y||. Prove that (b) More generally, if x and y are arbitrary nonzero vectors, let a the vector bx + ay bisects the angle between x and y. 21. Use vector methods to prove that the diagonals of a parallelogram bisect the vertex angles if and only if the parallelogram is a rhombus. 22. Given AABC with D on BC as shown in Figure 2.6. Prove that if AD bisects ZBAC, then BD|/CD| ||AB ||/|| A || . (Hint: Use Exercise 20b. Let x = AB and y = AC; give two expressions for AD in terms of x and y and use Exercise 1.1.10.)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Please help me solve problem 20. Thanks!

2 Dot Product 15 18. Use the Cauchy-Schwarz inequality to solve the following max/min problem: If the (long) diagonal of a rectangular box has length c, what is the greatest the sum of the length, width, and height of the box can be? For what shape box does the maximum occur? 19. Give an alternative proof of the Cauchy-Schwarz inequality, as follows. Let a = |x||, b = ||y||, and deduce from ||bx - ay l|20 that x y ab. Now how do you show that |x yl ab? When does equality hold? 20. (a) Letx and y be vectors with ||x || = lly|l. Prove that the vector x + y bisects the angle between x and y ||x || and b = ||y||. Prove that (b) More generally, if x and y are arbitrary nonzero vectors, let a the vector bx + ay bisects the angle between x and y. 21. Use vector methods to prove that the diagonals of a parallelogram bisect the vertex angles if and only if the parallelogram is a rhombus. 22. Given AABC with D on BC as shown in Figure 2.6. Prove that if AD bisects ZBAC, then BD|/CD| ||AB ||/|| A || . (Hint: Use Exercise 20b. Let x = AB and y = AC; give two expressions for AD in terms of x and y and use Exercise 1.1.10.)
Transcribed Image Text:2 Dot Product 15 18. Use the Cauchy-Schwarz inequality to solve the following max/min problem: If the (long) diagonal of a rectangular box has length c, what is the greatest the sum of the length, width, and height of the box can be? For what shape box does the maximum occur? 19. Give an alternative proof of the Cauchy-Schwarz inequality, as follows. Let a = |x||, b = ||y||, and deduce from ||bx - ay l|20 that x y ab. Now how do you show that |x yl ab? When does equality hold? 20. (a) Letx and y be vectors with ||x || = lly|l. Prove that the vector x + y bisects the angle between x and y ||x || and b = ||y||. Prove that (b) More generally, if x and y are arbitrary nonzero vectors, let a the vector bx + ay bisects the angle between x and y. 21. Use vector methods to prove that the diagonals of a parallelogram bisect the vertex angles if and only if the parallelogram is a rhombus. 22. Given AABC with D on BC as shown in Figure 2.6. Prove that if AD bisects ZBAC, then BD|/CD| ||AB ||/|| A || . (Hint: Use Exercise 20b. Let x = AB and y = AC; give two expressions for AD in terms of x and y and use Exercise 1.1.10.)
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