Sketching a Plane in Three Dimensional Space
Date: 11/15/2005 at 03:26:29 From: Peter Subject: Three simultaneous equations with three unknowns I know that an equation like ax + by + cz = p represents a plane in three dimensions. But I can't find any explanation on how to sketch such an equation, like 2x + y + z = 3. I'm particulary interested in what part the number 3 plays in the positioning of the plane on the xyz axes. Beyond sketching any one such equation, I'm also trying to sketch a system of simultaneous equations to interpret them geometrically. I would like to see clearly how three simultaneous equations interact with one another in the xyz axes. I am aware that three simultaneous equations may have a unique solution, be inconsistent (having no solution), or be dependent (having infinite solutions). But how do I sketch a system to show it clearly on the xyz axes?
Date: 11/15/2005 at 08:16:04 From: Doctor Rick Subject: Re: Three simultaneous equations with three unknowns Hi, Peter. It's really hard to sketch in three dimensions, since we have to do so on paper, making a 2-dimensional oblique view of the axes, and the third dimension can't be clearly distinguished. Here is one thing that may help you visualize the planes. Try plotting the points where a plane is intersected by each coordinate axis. For example, the plane defined by 2x + y + z = 3 crosses the x axis at (3/2,0,0) - I just set y = z = 0 and solved the resulting equation 2x = 3. Likewise it crosses the y axis at (0,3,0) and the z axis at (0,0,3). Draw the three axes, mark the three points (A, B, and C below), and connect them to form a triangle. z | C* | | | | | | | | * / \ A / \ B * * / \ / y / / / x This triangle lies in the plane; if you do the same for each plane, you'll get some idea of their relationships. For instance, you'll see if they are parallel. You may or may not see the line of intersection of two planes immediately, but if you extend the sides of the triangles until they intersect, you will. For instance, consider the lines of intersection of two planes with the x-y plane: these will be the lines joining the x-intercept and the y-intercept of each plane. Where they intersect is one point on the intersection of the planes. Do the same with the y-z plane (or the x-z plane), and connect the two points you have found to make the line of intersection. - Doctor Rick, The Math Forum http://mathforum.org/dr.math/
Date: 11/16/2005 at 16:02:45 From: Peter Subject: Thank you (Three simultaneous equations with three unknowns) Dear Doctor Rick - Thank you very much for your clear explanation and for your prompt reply. I appreciated it very much.
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