Find an equation of the plane.?

Question

The plane that passes through the point (−3, 1, 2) and contains the line of intersection of the planes x + y − z = 1 and 2x − y + 3z = 3.

Answer 1

Step 1:

The plane passes through the point .

The plane equations are

The cross product of two normal vectors is the direction vector for the line of intersection.

The normal vectors of two planes are

The cross product of two vectors is

The direction vector of line of intersection is .

Step 2:

Find the point lies in line of intersection, set .

Substitute in .

Substitute in .

Add  and .

Substitute in .

The point lies in line of intersection is .

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Contd...

Step 3:

The plane parallel to the line of intersection is .

Let be the vector from to .

The direction vector of line of intersection is .

The line of intersection is cross product of  and .

The normal vector is .

Substitute .

The plane equation is

The plane equation is .

Solution:

The plane equation is .