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feryee
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why do we consider normal line in expressing a plane,say in ##R^3## ,of the form ## ax + by + cz = d ##? What is the logic behind this normal line selection? Plz provide intuitive explanations.Thanks
The right hand rule might be a suitable illustration why it is useful.feryee said:why do we consider normal line in expressing a plane,say in ##R^3## ,of the form ## ax + by + cz = d ##? What is the logic behind this normal line selection? Plz provide intuitive explanations.Thanks
What exactly do you mean by convenient way? I couldn't still get it clearly? would you elaborate more on this!mfb said:It is a convenient way to express the position and orientation of the plane with a single equation.
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).
It is easy to use it in calculations.feryee said:What exactly do you mean by convenient way?
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?Mark44 said:For a plane in ##\mathbb{R}^3## whose equation is ax + by + cz = d, the vector ##\vec{N} = <a, b, c>## is perpendicular to (or normal to) the plane. The normal gives the orientation of the plane in the three-dimensional space, but that same vector is perpendicular to an infinite family of parallel planes. If you also know a point on the plane, that narrows the choices down to a single plane.
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.feryee said: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?
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)Mark44 said: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 P0, as in the drawing below.
Take an arbitrary point on the plane P(x, y, z) that is different from P0 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
I'm not sure which vectors you're talking about -- the normal or a vector in the plane.feryee said: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)
feryee said: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)
The normal line is a straight line that is perpendicular to the plane. It intersects the plane at a 90-degree angle and can be used to determine the orientation of the plane.
To find the normal line of a plane, you need to know the equation of the plane and the direction of its normal vector. The normal line can then be determined by finding a point on the plane and using the direction of the normal vector to find a second point on the line.
The normal line in expressing a plane is used to determine the orientation and direction of the plane. It is also useful in calculating the angle between two planes or determining the shortest distance from a point to a plane.
No, the normal line cannot be parallel to the plane because it is perpendicular to the plane. If the normal line and the plane were parallel, they would never intersect at a 90-degree angle.
The normal line and the concept of orthogonality are closely related because the normal line is always perpendicular to the plane, which is a key aspect of orthogonality. Additionally, the normal line can be used to determine if two planes are orthogonal to each other.