A plane is a smooth, two-dimensional surface, which stretches infinitely far. A plane is a two - dimensional representation of a point (zero dimensions), a line (one dimension) and a three-dimensional object. A plane in 3-dimensional space has the equation ax + by + cz + d = 0, where at least one of the coefficients a, b or c must be non-zero.
A vector is a physical quantity for which both direction and magnitude are defined. A position vector basically defines the position of a particular point in a three dimensional cartesian plane system, with respect to an origin point. Consider a line on a plane. This line has a length and an arrow. Here, the length is the magnitude and the arrowhead show the direction. Hence, in a plane, a line is a vector.
For one particular point on the vector, however, there is only one unique plane which passes through it and is also perpendicular to the vector. A vector can be thought of as a collection of points. So, for a particular vector, there are infinite planes which are perpendicular to it.
The vector equation for the following image is written as: (\[\overrightarrow{r}\] — \[\overrightarrow{r}_{0}\]). \[\overrightarrow{N}\] = 0, where \[\overrightarrow{r}\] and \[\overrightarrow{r}_{0}\] represent the position vectors. For this plane, the cartesian equation is written as:
A (x−x1) + B (y−y1) + C (z−z1) = 0, where A, B, and C are the direction ratios.
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P(x1, y1, z1), Q(x2, y2, z2), and R (x3, y3, z3) are three non-collinear points on a plane.
We know that: ax + by + cz + d = 0 —————(i)
By plugging in the values of the points P, Q, and R into equation (i), we get the following:
a(x1) + b(y1) + c(z1) + d = 0
a(x2) + b(y2) + c(z2) + d = 0
a(x3) + b(y3) + c(z3) + d = 0
Suppose, P = (1,0,2), Q = (2,1,1), and R = (−1,2,1)
Then, by substituting the values in the above equations, we get the following:
a(1) + b(0) + c(2) + d = 0
a(2) + b(1) + c(1) + d = 0
a(-1) + b(2) + c(1) + d = 0
Solving these equations gives us b = 3a, c = 4a, and d = (-9)a. ———————(ii)
By plugging in the values from (ii) into (i), we end up with the following:
ax + by + cz + d = 0
ax + 3ay + 4az−9a
x + 3y + 4z−9
Therefore, the equation of the plane with the three non-collinear points P, Q, and R is x + 3y + 4z−9.
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Example 1: A (3,1,2), B (6,1,2), and C (0,2,0) are three non-collinear points on a plane. Find the equation of the plane.
Solution:
We know that: ax + by + cz + d = 0 —————(i)
By plugging in the values of the points A, B, and C into equation (i), we get the following:
a(3) + b(1) + c(2) + d = 0
a(6) + b(1) + c(2) + d = 0
a(0) + b(2) + c(0) + d = 0
Solving these equations gives us a = 0, c = \[\frac{1}{2}\] b, d = —2b ———————(ii)
By plugging in the values from (ii) into (i), we end up with the following:
ax + by + cz + d = 0
0x + (—by) + \[\frac{1}{2}\] bz — 2b = 0
x - y + \[\frac{1}{2}\] z —2 = 0
2x-2y + z-4 = 0
Therefore, the equation of the plane with the three non-collinear points A, B and C is
2x-2y + z-4 = 0.
Example 2: S (0,0,2), U (1, 0, 1), and V (3, 1,1) are three non-collinear points on a plane. Find the equation of the plane.
Solution:
We know that: ax + by + cz + d = 0 —————(i)
By plugging in the values of the points S, U, and V into equation (i), we get the following:
a(0) + b(0) + c(2) + d = 0
a(1) + b(0) + c(1) + d = 0
a(3) + b(1) + c(1) + d = 0
Solving these equations gives us b = —2a, c = a, d = —2a ———————(ii)
By plugging in the values from (ii) into (i), we end up with the following:
ax + by + cz + d = 0
ax + —2ay + az — 2a = 0
x-2y + z-2 = 0
Therefore, the equation of the plane with the three non-collinear points A, B and C is
x-2y + z-2 = 0.
1. What is a Cartesian Plane?
The Cartesian plane, also known as the coordinate plane, is a two-dimensional plane generated by two perpendicular lines described as the x-axis (horizontal axis) and the y-axis (vertical axis). The exact position of the point on the Cartesian plane can be determined using coordinates that are written in the form of an ordered pair (x, y). Coordinates are a series of values that helps one to signify the exact position of a point in a coordinate plane. The distance of the point from the y-axis is called the abscissa. The distance of the point from the x-axis is called the ordinate.
2. What are Collinear and Non-Collinear Points?
Two or more points are said to be collinear if there is one line passing through all of them. Collinear points are connected by a line. Two points are always collinear, because the line connecting both of them is always present. Three points and above may or may not be collinear.
Non-collinear points are basically those points which do not lie on the same line.
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