Question

Find an equation for the plane that contains the point P, is parallel to A, and orthogonal to the plane you found in (c). Write all three forms of the equation for the plane.

A. Determine the equation of a plane based on given conditions.
B. Explore the equation of planes using points and vectors.
C. Explain the properties of a plane parallel to a given line and orthogonal to another plane.
D. Derive the equation of a plane based on specific requirements.

Answer 1

To find the equation for the plane that contains the point P, is parallel to A, and orthogonal to the given plane, we can use the equation of a plane in the form Ax + By + Cz = D. By finding the vector normal to the plane, we can substitute the coordinates of point P into the equation to solve for D. The equation of the plane can also be written in standard form and vector form.

Step-by-step explanation:

To find an equation for the plane that contains the point P and is parallel to vector A, we need to use the equation of a plane in the form Ax + By + Cz = D, where (x, y, z) are the coordinates of any point on the plane.

Since the plane is parallel to vector A, the vector normal to the plane will be perpendicular to A. We can find the vector normal to the plane by taking the cross product of the normal vectors of the two given planes in part (c).

Once we have the vector normal to the plane, we can substitute the coordinates of point P into the equation and solve for D. This will give us the equation of the plane in the form Ax + By + Cz = D.

To write the equation of the plane in the standard form (Ax + By + Cz + D = 0) and the vector form (r · n = D), we can rearrange the terms accordingly.