Non-Collinear Points - Equation of Plane
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 axe + 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 shows the direction. Hence, in a plane, a line is a vector.
Perpendicular Planes to Vectors and Points
For one particular point on the vector, however, there is only one unique plane that 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 that 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:
\[A (x - x_{1}) + B (y - y_{1}) + C (z - z_{1}) = 0\], where A, B, and C are the direction ratios.
Equation of Plane Passing through 3 Non-Collinear Points
\[P(x_{1},y_{1},z_{1}),~Q(x_{2},y_{2},z_{2}),~and~R(x_{3},y_{3},z_{3})\] 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:
\[a(x_{1}) + b(y_{1}) + c(z_{1}) + d = 0\]
\[a(x_{2}) + b(y_{2}) + c(z_{2}) + d = 0\]
\[a(x_{3}) + b(y_{3}) + c(z_{3}) + d = 0\]
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.
Solved Examples
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 =12
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) +12bz — 2b = 0
x - y +12z —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.
FAQs on Equation of a Plane Passing Through 3 Non Collinear Points
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 that do not lie on the same line. When there are more than three points, even if a single point lies outside the line then it becomes a collinear point.
3. How are non-collinear points different from collinear points?
Collinear points always lie on a straight line which need not necessarily exist as co-planar. They are a set of three or more points that may exist on different planes but not on different lines. Only one line is possible to pass through the three points. On the other hand, if these three points do not lie on the same straight line, they are called non-collinear points. Even if a point between two points lies outside the line, they are non-collinear points.
4. How do students practice questions on Non-Collinear points?
Because this chapter involves a graph, students must take extra caution. This chapter is especially tricky because often students are not very careful while marking the points on the graph. This is the reason why they should practice the graph many times to get a hang of the topic. Students can follow the NCERTs or RD Sharma for both easy and complex questions given after every chapter. Plus they should also check out the examples which provide many methods of solving similar questions.
5. Where can students get the sample papers to solve questions on Non-Collinear points?
For Maths, sample papers have to be solved to boost the confidence and to evaluate the progress of learning. Sample papers have been provided to the students to help them strengthen the foundation of different concepts and to have an idea about easy and complex questions which may be asked in the exam. Vedantu has curated, updated, and uploaded sample papers according to the latest syllabus. They are available on the official website which can be accessed by the students for free and can download the papers in PDF format.