# 1 a 2 3 1 b 2 0 3 2 a 4 2 1 b 2 2 1 3 a 2 4 0 b 1 1 2

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Chapter 7 / Exercise 2
Algebra and Trigonometry: Real Mathematics, Real People
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1. a = (2, 3, 1), b = ( 2, 0, 3). 2. a = (4, 2, 1), b = ( 2, 2, 1). 3. a = (2, 4, 0), b = (1, 1 2 , 0). 4. a = ( 2, 0, 5), b = (3, 0, 1). 5. a = 2 i + j 2 k , b = i + j + 2 k . 6. a = 2 i + 3 j + k , b = i + 4 j . Simplify. 7. (3 a b ) ( a 2 b ). 8. a ( a b ) + b ( b + a ). 9. ( a b ) c + b ( c + a ). 10. a ( a + 2 c ) + (2 b a ) ( a + 2 c ) 2 b ( a + 2 c ). 11. Taking a = 2 i + j , b = 3 i j + 2 k , c = 4 i + 3 k , calculate: (a) the three dot products a b , a c , b c ; (b) the cosines of the angles between these vectors; (c) the component of a (i) in the b direction, (ii) in the c direction; (d) the projection of a (i) in the b direction, (ii) in the c direction. 12. Repeat Exercise 11 with a = j + 3 k , b = 2 i j + 2 k , c = 3 i k . 13. Find the unit vector with direction angles 1 3 π , 1 4 π , 2 3 π . 14. Find the vector of norm 2 with direction angles 1 4 π , 1 4 π , 1 2 π . 15. Find the angle between the vectors 3 i j 2 k and i + 2 j 3 k . 16. Find the angle between the vectors 2 i 3 j + k and 3 i + j + 9 k . 17. Find the direction angles of the vector i j + 2 k . 18. Find the direction angles of the vector i 3 k . c Estimate the angle between the vectors. Express your answers in radians rounded to the nearest hundredth of a radian, and in degrees to the nearest tenth of a degree. 19. a = (3, 1, 1), b = ( 2, 1, 4). 20. a = ( 2, 3, 0), b = ( 6, 0, 4). 21. a = − i + 2 k , b = 3 i + 4 j 5 k . 22. a = − 3 i + j k , b = i j . c 23. Use a CAS to determine the angles and the perimeter of the triangle with vertices P (1, 3, 2), Q (3, 1, 2), R (2, 3, 1). c 24. Use a CAS to find the direction cosines and the direction angles of the vector from P (5, 7, 2) to Q ( 3, 4, 1). Find the direction cosines and direction angles of the vector. Express the angles in degrees rounded to the nearest tenth of a degree. 25. a = (1, 2, 2). 26. a = (2, 6, 1). 27. a = 3 i + 12 j + 4 k . 28. a = 3 i + 5 j 4 k . 29. Find all the numbers x for which 2 i + 5 j + 2 x k 6 i + 4 j x k . 30. Find all the numbers x for which ( x i + 11 j 3 k ) (2 x i x j 5 k ). 31. Find all the numbers x for which the angle between c = x i + j + k and d = i + x j + k is 1 3 π . 32. Set a = i + x j + k and b = 2 i j + y k . Compute all values of x and y for which a b and || a || = || b || . 33. (a) Show that 1 4 π , 1 6 π , 2 3 π cannot be the direction angles of a vector. (b) Show that, if a = a 1 i + a 2 j + a 3 k has direction angles α , 1 4 π , 1 4 π , then a 1 = 0. 34. If a vector has direction angles α = π/ 3, β = π/ 4, find the third direction angle γ . 35. What are the direction angles of a if the direction angles of a are α , β , γ ? 36. Suppose that the direction angles of a vector are equal. What are the angles? 37. Find the unit vectors u that are perpendicular to both i + 2 j + k and 3 i 4 j + 2 k . 38. Find two mutually perpendicular unit vectors that are perpendicular to 2 i + 3 j . 39. Find the angle between the diagonal of a cube and one of the edges. 40. Find the angle between the diagonal of a cube and the diagonal of one of the faces.
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The document you are viewing contains questions related to this textbook. The document you are viewing contains questions related to this textbook.
Chapter 7 / Exercise 2
Algebra and Trigonometry: Real Mathematics, Real People
Larson Expert Verified
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