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horizontal axes, we know that its slope is 1 Because we ve been told that line L passes through the origin, we know that its y-intercept is 0 From algebra, we remember that the slope-intercept form of the Cartesian equation for a straight line is y = mx + b where m is the slope and b is the y-intercept Plugging in 1 for m and 0 for b, we find that the Cartesian equation for line L is y = x We ve been told that circle C is centered at the origin and has a radius of 3 units From algebra, we recall that the general form for the equation of a circle centered at the origin is x2 + y2 = r2 where r is the radius When we plug in either 3 or 3 for r, we find that the Cartesian equation for circle C is x2 + y2 = 9 9 Here s the system of Cartesian equations that we ve found, representing line L and circle C as shown in Figs 3-8 and A-5: y = x and x 2 + y2 = 9 Let s replace y in the second equation by x, so we get x2 + ( x)2 = 9 Because ( x)2 = x2 for any real number x, we can rewrite the above equation as x2 + x2 = 9 which simplifies to 2x2 = 9 and further to x2 = 9/2 The solutions to this equation are x = (9/2)1/2
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480 Worked-Out Solutions to Exercises: 1-9
or x = (9/2)1/2 To solve for y, we must plug in these values of x to either of the equations in our original system Let s use y = x When we put the first of these solutions into that equation, we obtain y = (9/2)1/2 which tells us that one of the points is (x,y) = [(9/2)1/2, (9/2)1/2] When we plug the second solution for x into the equation y = x, we get y = [ (9/2)1/2] = (9/2)1/2 so we know that the other point is (x,y) = [ (9/2)1/2,(9/2)1/2] By inspecting Fig A-5, we can see that the points must be P = [ (9/2)1/2,(9/2)1/2] and Q = [(9/2)1/2, (9/2)1/2] 10 To get the Cartesian equivalents of the points we found when we solved Problems 6 and 7, we use the conversion formulas x = r cos q and y = r sin q The polar form of point P is (q,r) = (3p /4,3) In this case, we have x = 3 cos (3p /4) = 3 ( 21/2)/2 = (9/2)1/2 and y = 3 sin (3p /4) = 3 21/2/2 = (9/2)1/2 so the ordered pair is (x,y) = [ (9/2)1/2,(9/2)1/2]
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The polar form of point Q is (q,r) = (7p /4,3) In this case, we have x = 3 cos (7p /4) = 3 21/2/2 = (9/2)1/2 and y = 3 sin (7p /4) = 3 ( 21/2)/2 = (9/2)1/2 so the ordered pair is (x,y) = [(9/2)1/2, (9/2)1/2] We have found that P = [ (9/2)1/2,(9/2)1/2] and Q = [(9/2)1/2, (9/2)1/2] These results agree with what we got when we solved Problem 9 They are the Cartesian coordinates of points P and Q as shown in Figs 3-8 and A-5
4
1 Here are the two vectors we ve been told to work with: a = ( 3,6) and b = (2,5) In this situation, xa = 3, xb = 2, ya = 6, and yb = 5 The Cartesian sum a + b is a + b = [(xa + xb),( ya + yb)] = [( 3 + 2),(6 + 5)] = ( 1,11) Reversing the order of the sum, we get b + a = [(xb + xa),( yb + ya)] = [2 + ( 3),(5 + 6)] = ( 1,11)
482 Worked-Out Solutions to Exercises: 1-9
The Cartesian difference a - b is a - b = [(xa xb),( ya yb)] = [( 3 2),(6 5)] = ( 5,1) Reversing the order of the difference, we obtain b - a = [(xb xa),( yb ya)] = {[2 ( 3)],(5 6)} = [(2 + 3),(5 6)] = (5, 1) 2 Imagine that we have an arbitrary Cartesian vector a = (xa,ya) Its Cartesian negative is -a = ( xa, ya) By definition, the Cartesian sum vector a + ( a) is a + ( a) = {[xa + ( xa)],[ya + ( ya)]} = [(xa xa),( ya ya)] = (0,0) = 0 Reversing the order of the sum, we get a + a = [( xa + xa),( ya + ya)] = {[xa + ( xa)],[ya + ( ya)]} = [(xa xa),( ya ya)] = (0,0) = 0 3 As with the solutions to Problems 1 and 2, demonstrating this fact is a mere exercise in arithmetic Nevertheless, we can get some practice in mathematical rigor by carefully working our way through each step in the process According to the formula for the Cartesian difference between two vectors from the chapter text, we have a - b = [(xa xb),( ya yb)] and b - a = [(xb xa),( yb ya)] Now let s look closely at the coordinates for these two vectors, and compare them The x coordinate of a - b is the real number xa xb, while the x coordinate of b - a is the real number xb xa From pre-algebra, we remember that when we reverse the order of the difference between two numbers, we get the negative In this case, it means xb xa = (xa xb)
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