Final answer:
The equation of the plane that passes through the origin and the given points (5, -4, 2) and (1, 1, 1) is 4x - 5y + z = 0.
Step-by-step explanation:
To find the equation of the plane that passes through the origin and the points (5, -4, 2) and (1, 1, 1), we can use the formula:
ax + by + cz = d
where (x, y, z) are the coordinates of any point on the plane, and (a, b, c) are the components of the plane's normal vector.
Since the plane passes through the origin, we know that its normal vector is perpendicular to the line connecting the origin to any point on the plane. Therefore, we can find the normal vector by taking the cross product of the two vectors formed by subtracting the coordinates of the two given points from each other: (5-1, -4-1, 2-1) = (4, -5, 1).
So, the equation of the plane is 4x - 5y + z = 0.