A plane is defined by a point and two non-parallel vectors. Some linear sum of the vectors and the point will define any point in the plane. Any two vectors will do, hence there will be an infinite number of possible parametric equations for a plane.
The plane you have defined intersects the origin. One way to find a couple of vectors that lie in the plane would be to look at the vectors between axis intercept points. As this plane has no distinct intercepts, it is convenient to choose a parallel plane that does. Using the LCM of the coefficients, we can make use of the parallel plane
.. x - 5y - 7z = 35
.. x/35 - y/7 - z/5 = 1 ... a plane parallel to the given plane
The intercepts of this plane with the axes are (35, 0, 0), (0, -7, 0), (0, 0, -5). The vector from the y-intercept to the x intercept is (35, 7, 0) or (5, 1, 0). The vector from the z-intercept to the y-intercept is (0, -7, 5). We can use a multiplier of "a" on the first vector and "b" on the second vector to give the parametric equations for our plane in terms of a and b as
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A plane is defined by a point and two non-parallel vectors. Some linear sum of the vectors and the point will define any point in the plane. Any two vectors will do, hence there will be an infinite number of possible parametric equations for a plane.
The plane you have defined intersects the origin. One way to find a couple of vectors that lie in the plane would be to look at the vectors between axis intercept points. As this plane has no distinct intercepts, it is convenient to choose a parallel plane that does. Using the LCM of the coefficients, we can make use of the parallel plane
.. x - 5y - 7z = 35
.. x/35 - y/7 - z/5 = 1 ... a plane parallel to the given plane
The intercepts of this plane with the axes are (35, 0, 0), (0, -7, 0), (0, 0, -5). The vector from the y-intercept to the x intercept is (35, 7, 0) or (5, 1, 0). The vector from the z-intercept to the y-intercept is (0, -7, 5). We can use a multiplier of "a" on the first vector and "b" on the second vector to give the parametric equations for our plane in terms of a and b as
.. (x, y, z) = (0, 0, 0) + a(5, 1, 0) + b(0, -7, 5)
or, for some "a" and "b",
.. x = 5a
.. y = a - 7b
.. z = 5b
You need other relationships that match x with y or x with z, y with z