Final answer:
To find the equation of a plane given three points, one can use vector, scalar, or standard form equations. The vector form involves directional vectors from a fixed point, while the scalar and standard forms involve the plane's normal vector.
The correct answwer is d) All of the above
Step-by-step explanation:
To find the equation of the plane containing points mathbfa = (a_1, a_2, a_3), mathbfb = (b_1, b_2, b_3), and mathbfp = (p_1, p_2, p_3) we could use any of the provided forms depending on what we need the equation for. The vector form (option a) uses two directional vectors to describe how to move within the plane from a fixed point (mathbfa).
The scalar form (option b) gives a direct equation relating the x, y, z coordinates of any point on the plane. The standard form (option c) represents the plane as all points whose position vectors have a zero scalar product with a vector normal to the plane, mathbfn, when subtracted from mathbfa.
To derive the scalar or standard form, we need to find the normal of the plane by taking the cross product of (mathbfb - mathbfa) and (mathbfp - mathbfa), and then use the normal to find the scalar equation of the plane.
The correct answwer is d) All of the above