Chapter 10.3, Problem 6E

To determine

To fill in the blank

Answer to Problem 6E

The volume V of a $\underset{\_}{parallelepiped}$ with vector u, v and w as adjacent edges is $V=\left|u.\left(v\times w\right)\right|$

Explanation of Solution

Given information :  $u,vandw$ are adjacent edges with volume, $V=\left|u.\left(v\times w\right)\right|$

Concept used:

Concept of cross product

Explanation: 

Given volume, $V=\left|u.\left(v\times w\right)\right|$

From the geometric definition of the cross product, we know that its magnitude $\Vert v\times w\Vert$ is the area of the parallelogram base, and that the direction of the vector $v\times w$ is perpendicular to the base.

Also, $V=\left|u.\left(v\times w\right)\right|$ represents area of base times height

This is defined as volume of parallelepiped

Therefore,

Volume V of a $\underset{\_}{parallelepiped}$ with vector u, v and w as adjacent edges is $V=\left|u.\left(v\times w\right)\right|$