Final answer:
To find an equation of the plane passing through the origin and the points (3, -4, 6) and (5, 3, 1), we can use the formula for the equation of a plane in 3D space. The equation of the plane is 3x - 4y + 6z = 0.
Step-by-step explanation:
To find an equation of the plane passing through the origin and the given points, we can use the formula for the equation of a plane in 3D space. The formula is
Ax + By + Cz = D,
where A, B, C are the directional ratios of the plane and D is a constant. Since the plane passes through the origin, we have A(0) + B(0) + C(0) = D, which simplifies to D = 0. Therefore, the equation of the plane becomes Ax + By + Cz = 0.
To find the directional ratios A, B, C, we can use the points (3, -4, 6) and (5, 3, 1). We can pick any two points on the plane to find the directional ratios, so let's use these two points. Subtracting the coordinates of the origin from each point, we get (3 - 0, -4 - 0, 6 - 0) and (5 - 0, 3 - 0, 1 - 0) respectively. The directional ratios are therefore (3, -4, 6) and (5, 3, 1).
Substituting these values in the equation of the plane, we get 3x - 4y + 6z = 0 as the equation of the plane passing through the origin and the given points.