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- Thread starter feryee
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mfb

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There is no ambiguity in the selection apart from an overall scaling of the equation (you can multiply a,b,c,d by a non-zero constant without changing the plane).

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fresh_42

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The right hand rule might be a suitable illustration why it is useful.

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What exactly do you mean by convenient way? I couldn't still get it clearly? would you elaborate more on this!

There is no ambiguity in the selection apart from an overall scaling of the equation (you can multiply a,b,c,d by a non-zero constant without changing the plane).

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Mark44

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mfb

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It is easy to use it in calculations.What exactly do you mean by convenient way?

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Thank you very much. That explains quiet well. Can you please tell me how can one claim that this equation actually represent a plane in ##\mathbb{R}^3##? If i get your explanations right , the choice of d will determine if we have a single plane or infinite family of parallel plane. right?

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Mark44

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Think about it in terms of the geometry of the situtation. If you have a vector in ##\mathbb{R}^3##, it is perpendicular to an infinite number of planes, all with the same orientation (and therefore parallel). A given value of d identifies one and only one of these planes.Thank you very much. That explains quiet well. Can you please tell me how can one claim that this equation actually represent a plane in ##\mathbb{R}^3##?

If d is unknown, then yes, we have a family of parallel planes.feryee said:If i get your explanations right , the choice of d will determine if we have a single plane or infinite family of parallel plane. right?

For the algebra, let's say we know a vector N = <a, b, c> that is perpendicular to the plane, and a point ##P_0(x_0, y_0, z_0)## that is on the plane. Position the vector so that its tail is at P

Take an arbitrary point on the plane P(x, y, z) that is different from P

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Yeah i see. Really great. But when it comes to plotting the plane by hand, wouldn't this becomes difficult to find all of such a vector on the plane?How can the plotting done in the easy/ handy way(not using software)Think about it in terms of the geometry of the situtation. If you have a vector in ##\mathbb{R}^3##, it is perpendicular to an infinite number of planes, all with the same orientation (and therefore parallel). A given value of d identifies one and only one of these planes.

If d is unknown, then yes, we have a family of parallel planes.

For the algebra, let's say we know a vector N = <a, b, c> that is perpendicular to the plane, and a point ##P_0(x_0, y_0, z_0)## that is on the plane. Position the vector so that its tail is at P_{0}, as in the drawing below.

Take an arbitrary point on the plane P(x, y, z) that is different from P_{0}and form the vector ##\vec{P_0P} = <x - x_0, y - y_0, z - z_0>##. Since ##\vec{P_0P}## and ##\vec{N}## are perpendicular, their dot product must be zero. IOW, ##\vec{P_0P} \cdot \vec{N} = 0##, so ##(x - x_0) \cdot a + (y - y_0) \cdot b + (z - z_0) \cdot c = 0##. The equation ax + by + cz = d comes directly from this dot product.

View attachment 91587

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Mark44

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I'm not sure which vectors you're talking about -- the normal or a vector in the plane.Yeah i see. Really great. But when it comes to plotting the plane by hand, wouldn't this becomes difficult to find all of such a vector on the plane?

The question as you originally asked it didn't have anything to do with graphing a plane. If you have the equation of a plane, it's easy to find three points in the plane, and you can use these points to sketch the plane.feryee said:How can the plotting done in the easy/ handy way(not using software)

Or, you can find the intersection of the plane in the three coordinate planes.

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To plot by hand, it is generally much easier to plot where the plane intersects one of the 8 octants. For example, given ax+by+cz=d, set x=0 and you get by+cz=d. This produces a line that you can graph on the yz-plane. Similarly, set y=0 to get ax+cz=d -- a line in the xz-plane. Lastly, ax+by=d (when z=0) gives a line in the xy-plane. These three lines would be the edges of a triangle that lies in the plane ax+by+cz=d.Yeah i see. Really great. But when it comes to plotting the plane by hand, wouldn't this becomes difficult to find all of such a vector on the plane?How can the plotting done in the easy/ handy way(not using software)

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