Elementary Linear Algebra, First Canadian Edition by Stewart Venit, Wayne Bishop, Jason Brown

By Stewart Venit, Wayne Bishop, Jason Brown

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Sample text

Find an equation of the plane (in R3) that contairu the points (1, 0, 0), (0, 1, 0), and {0, 0, 1). 15. The equation of a plane in R3 is 3x I 2z = 1. Give a point-normal form for this y equation. 16. A plane (in R3) has equation {2, -1, 3) equation in standard form. o ((x, )I z) - ( -1, 3, 0)) - 0. Give this 17. Find an equation of the line in R 3 that completely contains the vector v - (1, 0, 2). + t(O, 1, 0) and 18. Determine the point (in R3) of intersection of the line x(t) - (1, 0, 1) the plane x I y I z; - 0.

29. - + j 0 x1 x4"5 =3 30. Find the midpoint of the line segment that joins (2, 1, 3, 5) with (6, 3. 2, 1). 31. Prove the properties ofTheorem 2. 32. The angle bt:twt:t:n two nonuro v«tors u and v in R"' is defined to he the unique number 9, 0 :s; 8 :s; 'If radians such that cos 8 Find the cosine of the angle between the vectors (3, 1, 1, 2, 1) and (0, 2, 1, 2, 0). CIIull llvll). 33. Prove the properties ofTheorem 3. 34. Describe the plane determined by the points (3, 1, 0, 2, 1), (2, 1, 4, 2, 0), and (-1, 2, 1, 3, 1).

33. Jx2 I y 2 I c1x I CzY I c3 = 0. Suppose the points (1, 1), (0, 2), and (k, 1) lie on the same circle. Solve for the equation of the circle when k = -1, 0, 1, 2, and 3. ) For what values of k is there no solution? 34. Use a CAS or a graphing calculator to plot the points and corresponding solution for the given values of kin Exercise 33. 35. Since a CAS can perform symbolic manipulation, solve directly for the: equation of the circle: with points ( 2, 0), (1, 3), (5, k). For what value of k is there no solution?