### Vector Equation of the Plane

To determine the equation of a plane in 3D space, a point P and a pair of vectors which form a basis (linearly independent vectors) must be known.

The point P belongs to the plane π if the vector is coplanar with the vectors and .

### Cartesian Equation of the Plane

A point is in the plane π if the system has the solution:

The values are given as:

### Intercept Form

A(a, 0, 0), B(0, b, 0) and C(0, 0, c).

### Examples

1.Find the equations of the plane that pass through point A = (1, 1, 1) and their direction vectors are: and .

2.Find the equations of the plane that pass through points A = (−1, 2, 3) and B = (3, 1, 4) and contains the vector .

3.Find the equations of the plane that pass through points A = (−1, 1, −1), B = (0, 1, 1) and C = (4, −3, 2).

4. π is the plane of parametric equations:

Confirm whether the points A = (2, 1, 9/2) and B = (0, 9, −1) belong to this plane.

5.Find the equation of the plane in intercept form that passes through the points A = (1, 1, 0), B = (1, 0, 1) and C = (0, 1, 1).

Divide by −2, and the equation is obtained:

6.Find the equation of the plane that passes through the point A = (2, 0, 1) and contains the line with the equation:

From the equation of the line, a second point and the vector is obtained.

7.Find the equation of the plane that passes through the points A = (1, −2, 4) and B = (0, 3, 2) and is parallel to the line:

8.Given the lines:

Determine the equation of the plane that contains the line r and is parallel to the line s.

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