When and how the store was started (lesson about = co-operate and prosper)
Answers
Answer:
Equation of a Plane<br>
A plane in 3-space has the equation<br>
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ax + by + cz = d,<br>
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where at least one of the numbers a, b, c must be nonzero.<br>
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As for the line, if the equation is multiplied by any nonzero constant k to get the equation kax + kby + kcz = kd, the plane of solutions is the same.<br>
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If c is not zero, it is often useful to think of the plane as the graph of a function z of x and y. The equation can be rearranged like this:<br>
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z = -(a/c)x + (-b/c) y + d/c<br>
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Another useful choice, when d is not zero, is to divide by d so that the constant term = 1.<br>
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(a/d)x + (b/d)y + (c/d)z = 1.<br>
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Another useful form of the equation is to divide by |(a,b,c)|, the square root of a2 + b2 + c2. This choice will be explained in the Normal Vector section.<br>
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Exercise: Where does the plane ax + by + cz = d intersect the coordinate axes?<br>
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Exercise: What is special about the equation of a plane that passes through 0.<br>
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Finding the equation of a plane through 3 points in space<br>
Given points P, Q, R in space, find the equation of the plane through the 3 points.<br>
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Example: P = (1, 1, 1), Q = (1, 2, 0), R = (-1, 2, 1). We seek the coefficients of an equation ax + by + cz = d, where P, Q and R satisfy the equations, thus:<br>
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a + b + c = d<br>
a + 2b + 0c = d<br>
-a + 2b + c = d<br>
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Subtracting the first equation from the second and then adding the first equation to the third, we eliminate a to get<br>
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b - c = 0<br>
4b + c = 2d<br>
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Adding the equations gives 5b = 2d, or b = (2/5)d, then solving for c = b = (2/5)d and then a = d - b - c = (1/5)d.<br>
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So the equation (with a nonzero constant left in to choose) is d(1/5)x + d(2/5)y + d(2/5)z = d, so one choice of constant gives<br>
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x + 2y + 2z = 5<br>
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or another choice would be (1/5)x + (2/5)y + (2/5)z = 1<br>
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Given the coordinates of P, Q, R, there is a formula for the coefficients of the plane that uses determinants or cross product.<br>
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Exercise. What is equation of the plane through the points I, J, K?<br>
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Exercise: What is the equation of the plane through (1, 1, 1), (-1, 1, -1), and (1, -1,