In the vector equation R = A + λ B + μ C of a plane, which statements are true about A, B, C?

In the vector equation R = A + λ B + μ C of a plane, which statements are true about A, B, C?

• A. A is a direction vector; B and C are fixed points
• B. A is the normal vector to the plane; B and C are direction vectors
• C. A is a fixed point on the plane; B and C are direction vectors
• D. A, B, C are all points on the plane

Answer: (C)

The main idea is that a plane is generated from a fixed point and two directions within the plane. In the equation, R represents any point on the plane, obtained by starting at a fixed point A and then moving along two independent directions B and C, scaled by λ and μ. This means A is a fixed point on the plane, and B and C are direction vectors that lie in the plane (they’re not points themselves, and they must not be parallel so they span the plane). The normal to the plane would be perpendicular to both B and C (given by B × C), but that isn’t part of this representation. So the statement that A is a fixed point on the plane and B and C are direction vectors is the correct interpretation.

The main idea is that a plane is generated from a fixed point and two directions within the plane. In the equation, R represents any point on the plane, obtained by starting at a fixed point A and then moving along two independent directions B and C, scaled by λ and μ. This means A is a fixed point on the plane, and B and C are direction vectors that lie in the plane (they’re not points themselves, and they must not be parallel so they span the plane). The normal to the plane would be perpendicular to both B and C (given by B × C), but that isn’t part of this representation. So the statement that A is a fixed point on the plane and B and C are direction vectors is the correct interpretation.