Final answer:
To find the equation of the plane through the origin and the points (2, −1, 5) and (8, 3, 3), we need to find the normal vector of the plane. The normal vector is perpendicular to any vector lying in the plane.
Step-by-step explanation:
To find the equation of the plane through the origin and the points (2, -1, 5) and (8, 3, 3), we need to find the normal vector of the plane. The normal vector is perpendicular to any vector lying in the plane. We can find the normal vector by taking the cross product of the vectors formed by subtracting the origin from the two given points.
Let's calculate the normal vector:
v1 = (2, -1, 5) - (0, 0, 0) = (2, -1, 5)
v2 = (8, 3, 3) - (0, 0, 0) = (8, 3, 3)
Normal vector = v1 x v2
Now, we can use the normal vector and the coordinates of the origin (0, 0, 0) to write the equation of the plane in the form ax + by + cz = d. Substituting the known values, we get:
2x - y + 4z = 0