Plane

Volume of Parallelepiped Determined by Vectors

Let be the plane defined by the equation:

Let be the set of vectors and the distance of from the plane is given as .

Let and be three distinct vectors in such that:

The quantity represents the volume of the parallelepiped determined by the vectors . The value of is given as:

The distance of a point from a plane is given by the formula:

In this case, the point is and the plane equation is . The distance is given as , so we have:

Simplifying the denominator:

This leads to the equation:

Next, since are three distinct vectors in , and the distance between the vectors is equal, we can conclude that these vectors form an equilateral triangle. The volume of the parallelepiped formed by three vectors is given by the scalar triple product:

Since , the vectors form an equilateral triangle, and we can calculate the volume of the parallelepiped using geometric properties of the vectors and the given distance. The final result for the volume is:

The value of is 45.