How to get the equation of the plane. [read this one]

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In summary, the equation of a plane is a representation of the relationship between x, y, and z coordinates and is typically written as ax + by + cz + d = 0. It can be found using three points or a normal vector and the constant term, represented by d, determines the distance of the plane from the origin. In real-life applications, it is used in fields such as engineering, aviation, and surveying to describe the orientation and position of objects in three-dimensional space.
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http://www.isquaredonline.com/chips/1.jpg

the above link is the answer to the problem I'm solving. i was wondering how i'd come up with the equation of the plane "x-y+z=0" when only the vertices of the plane are provided in the problem. thanks in advance.
 
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The equation of a plane through the origin is always of the form ax+by+cz=0.
Now you can plug in some points. Say, put in (1,1,0) and you'll find a=-b.
Putting in (0,1,1) b=-c. So the equation reduces to ax-ay+az=0 or x-y+z=0.
 
  • #3
thanks for the help.
 

1. What is the equation of a plane?

The equation of a plane is a mathematical representation that describes the relationship between the x, y, and z coordinates of points on the plane. It is typically written in the form ax + by + cz + d = 0, where a, b, and c are the coefficients of the variables and d is a constant term.

2. How do you find the equation of a plane given three points?

To find the equation of a plane given three points, you can use the formula (x-x1)(y-y1)(z-z1) = (x2-x1)(y2-y1)(z2-z1), where (x1, y1, z1), (x2, y2, z2), and (x3, y3, z3) are the three points. From this, you can rearrange the equation to get it in the form ax + by + cz + d = 0, where a, b, and c are the coefficients and d is a constant term.

3. Can you get the equation of a plane from a normal vector?

Yes, the equation of a plane can be written in the form ax + by + cz + d = 0, where (a, b, c) is the normal vector of the plane. To get the equation, you can choose any point on the plane and substitute its x, y, and z coordinates into the equation.

4. What is the significance of the constant term in the equation of a plane?

The constant term in the equation of a plane, represented by the letter d, determines the distance of the plane from the origin. This distance is measured along the direction of the normal vector. For example, if d = 0, the plane passes through the origin, and if d > 0, the plane is located on the side of the normal vector.

5. How is the equation of a plane used in real-life applications?

The equation of a plane has various real-life applications, including in engineering, aviation, and surveying. It is used to describe the orientation and position of objects in three-dimensional space, such as the angle of an airplane's flight path or the slope of a roof. It is also used in computer graphics to create 3D models and in physics to calculate the forces acting on an object.

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