What is the volume of the parallelepiped defined by the vectors ?
Volume Ex. 1

What is the volume of the tetrahedron defined by the vectors ?

To find the volume of the parallelepiped defined by the vectors , you can calculate the scalar triple product, which can be computed as the absolute value of the determinant of the matrix formed by these vectors:

Calculating the determinant gives:

Thus, the volume of the parallelepiped is cubic units.

For the volume of the tetrahedron defined by the same vectors, the formula is simply one-sixth of the volume of the parallelepiped. Hence, you can calculate the tetrahedron’s volume as follows:

So, the volume of the tetrahedron defined by those vectors is cubic units.

Now that you have your answers, here’s a bit of fun knowledge:

Did you know that parallelepipeds are like the 3D version of parallelograms? Just as a parallelogram derives its area from its base and height, a parallelepiped utilizes the same core idea but works in an extra dimension—where every face can be a parallelogram in its own right!

For those captivated by shapes and volumes, the study of polyhedra can lead you down a fascinating path. Books like “Geometry Revisited” by H. S. M. Coxeter and “Flatland” by Edwin A. Abbott can open your eyes to a world where dimensions come alive, packed with mathematical prowess and imagination!