Math Problem Statement

Find the equation of the plane through the point P=(3,3,2)P=(3,3,2) and parallel to the plane 5x−5y−z=−6.

Solution

The equation of a plane parallel to another plane has the same normal vector as the given plane. The general equation of a plane is:

Ax+By+Cz=DAx + By + Cz = D

Where AA, BB, and CC are the components of the normal vector.

Step 1: Identify the normal vector of the given plane

The equation of the given plane is:

5x5yz=65x - 5y - z = -6

The normal vector n\mathbf{n} of this plane is the coefficients of xx, yy, and zz, which gives:

n=(5,5,1)\mathbf{n} = (5, -5, -1)

Step 2: Write the general equation of the plane parallel to the given one

Since the plane we're looking for is parallel to the given plane, it will have the same normal vector, so its equation will have the form:

5x5yz=D5x - 5y - z = D

Step 3: Use the point P(3,3,2)P(3, 3, 2) to find DD

Substitute the coordinates of point P(3,3,2)P(3, 3, 2) into the equation to find DD:

5(3)5(3)(2)=D5(3) - 5(3) - (2) = D

15152=D15 - 15 - 2 = D

D=2D = -2

Step 4: Final equation of the plane

Therefore, the equation of the plane through the point P(3,3,2)P(3, 3, 2) and parallel to the plane 5x5yz=65x - 5y - z = -6 is:

5x5yz=25x - 5y - z = -2

Would you like more details on this or have any further questions?


Here are 5 related questions you might consider:

  1. How do you determine the normal vector from a plane equation?
  2. What is the geometric interpretation of parallel planes?
  3. Can you find the distance between two parallel planes?
  4. How do you find the equation of a plane passing through three points?
  5. What is the equation of a plane perpendicular to a given vector?

Tip: The constant DD in the equation of a plane can be quickly determined by substituting any known point on the plane into the equation.

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Math Problem Analysis

Mathematical Concepts

Analytic Geometry
Planes
Vectors

Formulas

Ax + By + Cz = D (Equation of a plane)
Substituting a point into the plane equation to find D

Theorems

Planes parallel to each other share the same normal vector

Suitable Grade Level

Grades 10-12