Q. 16 - Calculus

Question

What is a parallelepiped? What is meant by the parallelepiped determined by the vectors u, v and w? How do you find the volume of the  parallelepiped determined by u, v and w?

Step-by-Step Solution

Answer

A parallelepiped is a three-dimensional analog of a parallelogram, in much the same way that a cube is a three-dimensional analog of a square.

The parallelepiped determined by the vectors u, v and w  if they do not lie in the same plane.

$\mathrm{The} \mathrm{volume} \mathrm{of} \mathrm{the} \mathrm{parallelepiped}=|u·(v\times w\left)\right|$

Step 1. Given Information

What is a parallelepiped? What is meant by the parallelepiped determined by the vectors u, v and w? How do you find the volume of the parallelepiped determined by u, v and w?

Step 2. The Parallelepiped

A parallelepiped is a three-dimensional analog of a parallelogram, in much the same way that a cube is a three-dimensional analog of a square. 

Specifically, a parallelepiped is a six-sided solid whose surface consists of three pairs of parallel faces, each of which is a parallelogram.

Step 3. The parallelepiped determined by the vectors u, v and w

Any three vectors u, v and w in ${\mathrm{ℝ}}^3$ that do not lie in the same plane will

determine a parallelepiped.

Step 4. The volume of the parallelepiped

The area of a parallelogram involves a cross product. Similarly, the volume of the parallelepiped involves a triple scalar product.

$\mathrm{The} \mathrm{volume} \mathrm{of} \mathrm{the} \mathrm{parallelepiped}=u·(v\times w)$

if and only if u, v and w form a right-handed triple, and in any case the volume is$|u·(v\times w\left)\right|$.