Scalar product forms of a plane

Scalar product forms of a plane

Definition of a Plane

• A plane in three-dimensional space can be defined by an equation of the form ax + by + cz = d, where ‘a’, ‘b’ and ‘c’ are the components of a vector normal (perpendicular) to the plane, and ‘d’ is a constant.
• This equation can also be written in normal form as r.n = d, where r represents a position vector drawn from the origin to any point in the plane, and n (a, b, c) is the normal vector.

Vectors on a Plane

• Any two vectors in the plane which are not parallel can be used to form two direction vectors for the plane.
• These two vectors can be combined linearly (using scalar magnitude and vector addition) to determine any point in the plane.

Scalar Product Form

• An alternative way to represent a plane is using the scalar (dot) product. Given a point A with position vector a in the plane, and vectors b and c also lying in the plane, any point P with position vector r in the plane can be represented as: r = a + λb + µc, where λ and µ are scalars.
• The values of λ and µ can be varied to represent all possible position vectors r of points in the plane.

Intersection of a Line and a Plane

• If a line intersects a plane, the position vector r of any point on that line can be equated with the scalar product form of the plane to solve for the coordinates of the intersection point.
• This involves equating the line equation r = a + t(n) and the plane equation r = a + λb + µc, where n is the direction vector of the line.

Parallel and Perpendicular Planes

• Two planes are parallel if their normal vectors are parallel (i.e., they are scalar multiples of each other).
• Two planes are perpendicular if their normal vectors are perpendicular (i.e., their dot product equals zero).
• In both cases, these conditions can be deduced from the general equation of a plane and its scalar product form.

Distance from a Point to a Plane

• The distance from a point to a plane is the shortest distance between the point and any point in the plane. It is given by the absolute value of the scalar product of the normal vector to the plane and the vector formed between any point in the plane and the given point, divided by the magnitude of the normal vector.