This is the same as y = 5, and your three given points satisfy that equation. If you want to convince yourself absolutely, form vectors AB and AC, and dot them with the normal to your plane, which happens to be <0, 5, 0> (or any multiple of this vector).NATURE.M said:Homework Statement
Find the vector, parametric and scalar equation of the plane passing through A(3, 5, 2), B(0, 5, −1) and C(1, 5, −3).
Homework Equations
Vector, scalar, and parametric equations of plane.
The Attempt at a Solution
For my solution:
vector equation:
[x,y,z]=[3,5,2]+s[-3,0,-3]+t[-2,0,-5], s,t εℝ (from this the parametric equations are quite intuitive.
i really would like to confirm the scalar equation i have for the plane:
-9y+45=0
Mark44 said:This is the same as y = 5, and your three given points satisfy that equation. If you want to convince yourself absolutely, form vectors AB and AC, and dot them with the normal to your plane, which happens to be <0, 5, 0> (or any multiple of this vector).
Finding equations of a plane is important in various fields of science and engineering, such as physics, aerodynamics, and structural analysis. It allows us to understand the relationship between different points and lines on a 3D plane, and can be used to solve problems involving surfaces, intersections, and distances.
The equations of a plane can be determined using different methods, such as using the coordinates of three non-collinear points on the plane, or by using the normal vector and a point on the plane. The equations can also be derived from the general equation of a plane, which is ax + by + cz + d = 0, where a, b, and c represent the coefficients of the plane's normal vector, and d represents the distance of the plane from the origin.
Yes, equations of a plane can be found in any coordinate system, as long as the points and vectors used are consistent with the chosen coordinate system. For example, in a Cartesian coordinate system, the equations would involve x, y, and z coordinates, while in a polar coordinate system, the equations would involve r and θ coordinates.
There are some limitations when finding equations of a plane, such as the requirement of non-collinear points for certain methods, and the fact that the equations may not represent all points on the plane, but rather a specific region. Additionally, if the points used are not precise or there are errors in calculations, the resulting equations may not accurately represent the plane.
The equations of a plane have various real-world applications, such as in navigation systems, flight simulations, and architectural and engineering designs. They can also be used to analyze the stability and stress distribution of structures, and to determine the path of an object in 3D space. Additionally, the equations can be used in computer graphics to render 3D objects and environments.