Finding an equation of the plane (Linear Algebra)

In summary, to find an equation of a plane parallel to the plane containing points P, Q, and R, the directional vectors can be found using the given points and the cross product of these vectors can be used to find the normal vector. The equation of the plane can then be written in the form ax + by + cz + d = 0, with the y-intercept of the plane being (0, -5, 0).
  • #1
SmellyGoomba
2
0

Homework Statement


Find an equation of the plane that has y-intercept -5 and is parallel to the plane containing the points P(3, -1, 2), Q(0, 2, 1) and R(5, 2, 0)

Homework Equations



ax + by + cz + d = 0

The Attempt at a Solution



I got two directional vectors
u = PQ = (-3, 3, -1)
v = PR = (2, 3, -2)

n = u x v = (-3, -8, -15) which is also equal to (3, 8, 15) because it's a nonzero multiple of n

So putting P and the norm together, I get
3(x-3) + 8(y+1) + 15(z-2) = 0
3x + 8y + 15z - 31 = 0

Now I'm stuck. I have no idea what to do with y-intercept of -5. A push in the right direction would be nice.
 
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  • #2
The point P is not in your plane. It is in a parallel plane. The y-intercept (0, -5, 0) is the only point that is presented as being in your plane.
 

Related to Finding an equation of the plane (Linear Algebra)

1. What is an equation of the plane?

An equation of the plane is a mathematical representation that describes a flat, two-dimensional surface in three-dimensional space. It is usually written in the form of Ax + By + Cz + D = 0, where A, B, and C are constants and x, y, and z are variables.

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

To find the equation of a plane, you need to have at least three points on the plane or two non-parallel vectors that lie on the plane. Then, you can use these points or vectors to determine the values of A, B, C, and D in the equation Ax + By + Cz + D = 0. Another method is to use the normal vector of the plane and a point on the plane to find the equation.

3. What is the importance of finding the equation of a plane?

Finding the equation of a plane is important in many applications, such as computer graphics, engineering, and physics. It allows us to understand and visualize the properties of a plane, including its orientation, distance from the origin, and intersection with other planes or objects.

4. Can a plane have multiple equations?

Yes, a plane can have multiple equations. In fact, any equation that satisfies the condition of being a flat, two-dimensional surface in three-dimensional space can be considered an equation of the plane. However, we often prefer the simplest form of the equation, which has the lowest possible coefficients A, B, and C.

5. How does linear algebra relate to finding the equation of a plane?

Linear algebra is closely related to finding the equation of a plane because it deals with vectors and planes in three-dimensional space. The concept of linear independence and span of vectors is used to determine if the given points or vectors lie on the same plane. Also, matrix operations, such as row reduction, are often used to solve systems of equations and find the coefficients of the plane's equation.

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