HH_12_4 - Math 10C Lecture Examples Section 12.4 Linear...

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(10/8/08) Math 10C. Lecture Examples. Section 12.4. Linear functions Theorem (a) (The slope-intercept equation of a plane) Suppose that the z -intercept of a plane is b , the slope of its vertical cross sections in the positive x -direction is m 1 , and the slope of its vertical cross sections in the positive y -direction is m 2 (Figure 1). Then the plane has the equation, z = m 1 x + m 2 y + b . (1) (b) (The point-slope equation of a plane) Suppose that a plane contains the point ( x 0 , y 0 , z 0 ) , the slope of its vertical cross sections in the positive x -direction is m 1 , and the slope of its vertical cross sections in the positive y -direction is m 2 (Figure 2). Then the plane has the equation, z = z 0 + m 1 ( x - x 0 ) + m 2 ( y - y 0 ) . (2) The slope-intercept equation The point-slope equation FIGURE 1 FIGURE 2 Example 1 Give an equation of the plane with slope - 6 in the positive x -direction, slope 7 in the positive y -direction, and z -intercept 10. Answer: z = - 6 x + 7 y + 10 Example 2 Give an equation of the plane through the point (1,2,3) with slope 4 in the positive x -direction and slope - 5 in the positive y -direction. Answer: z = 3 + 4(
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