math125-lecture 2.4

math125-lecture 2.4 - Sec 2.4 Friday, January 29, 2010...

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Sec 2.4 Friday, January 29, 2010 11:01 PM Section2dot4 Page 1
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The first octant is the portion of R 3 defined by x ≥ 0, y ≥ 0, and z ≥ 0 2.4.1 Wednesday, February 03, 2010 3:12 PM Section2dot4 Page 2
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The first quadrant is the portion of R3 defined by x ≥ 0, y ≥ 0, and z ≥ 0 The equation of a plane in standard form is Ax + By + Cz = D where A, B, C, and D are constants 2.4.2 Wednesday, February 03, 2010 3:12 PM Section2dot4 Page 3
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The first quadrant is the portion of R3 defined by x ≥ 0, y ≥ 0, and z ≥ 0 The equation of a plane in standard form is Ax + By + Cz = D where A, B, C, and D are constants EX. Ax + By + Cz = 0 so (0, 0, 0) lies on this plane 2.4.3 Wednesday, February 03, 2010 3:12 PM Section2dot4 Page 4
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The first quadrant is the portion of R3 defined by x ≥ 0, y ≥ 0, and z ≥ 0 The equation of a plane in standard form is Ax + By + Cz = D where A, B, C, and D are constants EX. Ax + By + Cz = 0 so (0, 0, 0) lies on this plane Given that Ax + By + Cz = D and D ≠ 0, the intercept form of the equation of the plane is A/D x + B/D y + C/D z = 1 that is x/a + y/b + z/c = 1 where a = D/A, b = D/B, c = D/C, and assuming that A ≠ 0, B ≠ 0, C ≠ 0 2.4.4 Wednesday, February 03, 2010 3:12 PM Section2dot4 Page 5
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4 x + 8 y + 2 z = 16 (/16) 4 x/ 16 + 8 y/ 16 + 2 z /16 = 1 So the intercept form is x /4 + y /2 + z /8 = 1 2.4.5 Wednesday, February 03, 2010 3:21 PM Section2dot4 Page 6
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4 x + 8 y + 2 z = 16 (/16) 4 x/ 16 + 8 y/ 16 + 2 z /16 = 1 So the intercept form is x /4 + y /2 + z /8 = 1 x - intercept happens when y = 0 and z = 0 x /4 + 0/2 + 0/8 = 1 x = 4 so the x -intercept is (4, 0, 0) 2.4.6 Wednesday, February 03, 2010 3:21 PM Section2dot4 Page 7
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4 x + 8 y + 2 z = 16 (/16) 4 x/ 16 + 8 y/ 16 + 2 z /16 = 1 So the intercept form is x /4 + y /2 + z /8 = 1 x - intercept happens when y = 0 and z = 0 x /4 + 0/2 + 0/8 = 1 x = 4 so the x -intercept is (4, 0, 0) y - intercept happens when x = 0 and z = 0 0/4 + y /2 + 0/8 = 1 y = 2 so the y -intercept is (0, 2, 0) Similarly, the z - intercept is (0, 0, 8) 2.4.7 Wednesday, February 03, 2010 3:21 PM Section2dot4 Page 8
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4 x + 8 y + 2 z = 16 (/16) 4 x/ 16 + 8 y/ 16 + 2 z /16 = 1 So the intercept form is x /4 + y /2 + z /8 = 1 x - intercept happens when y = 0 and z = 0 x /4 + 0/2 + 0/8 = 1 x = 4 so the x -intercept is (4, 0, 0) y - intercept happens when x = 0 and z = 0 0/4 + y /2 + 0/8 = 1 y = 2 so the y -intercept is (0, 2, 0) Similarly, the z - intercept is (0, 0, 8) first octant portion of the plane 2.4.8 Wednesday, February 03, 2010 3:21 PM Section2dot4 Page 9
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i.e. 5(0) + 3(0) - 3(0) = 0 so (0, 0, 0) is on the plane. But the origin is the only intercept, so we need a new method for plotting
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math125-lecture 2.4 - Sec 2.4 Friday, January 29, 2010...

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