Final answer:
To find the equation of a plane passing through a point (1, 2, 3) and containing a line, the point-normal form is used along with the line's direction vector (2, 3, 5) as the normal vector to the plane.
Step-by-step explanation:
The student is asking about finding the equation of a plane that passes through a given point and lies along a given line. The given point is (1, 2, 3), and the line is given in parametric form as x/2 = (y-3)/3 = (z-1)/5. To find the equation of the plane, we can use the point-normal form of a plane's equation which is (x - x1)*n1 + (y - y1)*n2 + (z - z1)*n3 = 0, where (x1, y1, z1) is a point on the plane, and (n1, n2, n3) are the components of the normal vector to the plane.
First, we establish a direction vector for the line which can be used as a normal vector for the plane, since the plane is supposed to contain the line. We can read the direction vector for the line directly from the parametric equations of the line as (2, 3, 5). Next, we use the given point on the plane (1, 2, 3) as the point (x1, y1, z1). Finally, we substitute these values into the point-normal form to obtain the equation of the plane.