Understanding the Triple Scalar Product and Its Relationship to Volumes and Planes

Introduction to Triple Scalar Product and Vector Equations

Let's start by clarifying the distinction between an expression and an equation. An expression such as a triple scalar product is a mathematical entity that we manipulate or evaluate, whereas an equation makes a statement asserting the equality of two expressions. In the context of this discussion, we will focus on how these concepts relate to the orientation and positioning of planes and the volume of parallelepipeds.

Defining a Plane and the Vector Equation of a Plane

The equation of a plane in three-dimensional space specifies the spatial relationship between points and distances. A plane can be defined by two key pieces of information: 1) a point that lies on the plane, and 2) a vector that is normal (perpendicular) to the plane. Any equation that contains these two pieces of information describes the plane's position and orientation in space.

For a vector ( vec{A} ) and another vector ( vec{B} ) which lie in that plane, the cross product ( vec{A} times vec{B} ) gives a vector that is perpendicular to the plane. This normal vector is a key component in expressing the equation of the plane: If ( vec{P} - vec{Q} ) represents the vector from a known point ( vec{Q} ) to an arbitrary point ( vec{P} ) in the plane, then the equation of the plane is given by ( (vec{P} - vec{Q}) cdot (vec{A} times vec{B}) 0 ).

The Role of the Triple Scalar Product in Determining Volumes

The triple scalar product is another way to describe the volume of a parallelepiped. A parallelepiped is a three-dimensional figure formed by the vectors ( vec{A} ), ( vec{B} ), and ( vec{C} ). The volume of this parallelepiped can be determined by the absolute value of the triple scalar product: ((vec{A} cdot (vec{B} times vec{C}))).

Note that if ( vec{P} - vec{Q} ) is a vector lying in the plane defined by ( vec{A} ) and ( vec{B} ), then the triple scalar product expression ((vec{P} - vec{Q}) cdot (vec{A} times vec{B})) evaluates to zero because it represents the volume of a degenerate parallelepiped (a flat parallelogram), which has no volume.

Conclusion: Triple Scalar Product and the Vector Equation of a Plane

To summarize, the triple scalar product ((vec{P} - vec{Q}) cdot (vec{A} times vec{B})) does not equal the vector equation of the plane ((vec{P} - vec{Q}) cdot (vec{A} times vec{B}) 0). Instead, its absolute value represents the volume of a parallelepiped formed by the vectors (vec{A}), (vec{B}), and another vector orthogonal to the plane (such as (vec{P} - vec{Q}) when (vec{P}) is on the plane).

This understanding bridges the concepts of plane equations, vector products, and volume calculations, offering a powerful tool for solving problems in geometry and linear algebra.