Question

Give an equation for the plane parallel to the plane 2x + 2y + z = 5 and that contains the line with parametric equations x = 2 - t, y = 2t + 1, z = 3 - 2t

Answer 1

To find the equation for the plane parallel to the given plane and containing the given line, we need to find a normal vector to the given plane. The normal vector is the coefficients of x, y, and z in the plane equation. So, the equation for the desired plane is 2x + 2y + z = 13.

Step-by-step explanation:

To find the equation for the plane parallel to the given plane and containing the given line, we need to find a normal vector to the given plane. The normal vector is the coefficients of x, y, and z in the plane equation. So, the normal vector for the given plane is (2, 2, 1).

Now, since the desired plane is parallel to the given plane, the normal vector for the desired plane will be the same as the normal vector for the given plane. So, the equation for the desired plane will be of the form 2x + 2y + z = d, where d is a constant to be determined.

To find the value of d, we substitute the coordinates of a point on the given line into the equation of the desired plane. Let's use the point (2, 1, 3) on the given line:

2(2) + 2(1) + (3) = d

Simplifying the equation gives us: 8 + + 3 = d

d = 13

Therefore, the equation for the desired plane is 2x + 2y + z = 13.