Find the equation of a plane containing the two parallel lines v 1 = (0, 1, -2) + t(2,3, -1) and v 2 = (2, -1, 0) + t(2, 3, -1)

The equation of a plane in 3D space is given by the formula Ax + By + Cz = D, where A, B, C are the coefficients of the normal vector to the plane, and D is a constant.

The normal vector to the plane can be found by taking the cross product of the direction vectors of the two lines. However, in this case, the two lines are parallel, which means they have the same direction vector, (2, 3, -1). Therefore, we cannot find a unique normal vector using these two lines.

However, we know that the plane contains both lines, so it must contain the points (0, 1, -2) from line v1 and (2, -1, 0) from line v2. We can use these two points to find a vector in the plane, and then cross this with the direction vector to find the normal vector.

The vector between the points (0, 1, -2) and (2, -1, 0) is (2 - 0, -1 - 1, 0 - (-2)) = (2, -2, 2).

Now we can find the normal vector by taking the cross product of this vector and the direction vector (2, 3, -1):

(2, -2, 2) x (2, 3, -1) = (-2*-1 - 23, 22 - 2*-1, 23 - 2-2) = (2 - 6, 4 - (-2), 6 - (-4)) = (-4, 6, 10)

So the normal vector to the plane is (-4, 6, 10), and the equation of the plane is -4x + 6y + 10z = D.

To find D, we can substitute one of the points from the lines into the equation. Using the point (0, 1, -2) from line v1, we get:

-40 + 61 + 10*-2 = D D = 6 - 20 = -14

So the equation of the plane is -4x + 6y + 10z = -14.