Chapter 11.3, Problem 68E

To determine

To explain: how one could find the surface area of the parallelepiped.

Answer to Problem 68E

The vector $u=\langle 4,-4,2\rangle ,v=\langle 0,4,-3\rangle$ and $w=\langle 1,3,4\rangle$ can be used to find the surface $S$ .

Explanation of Solution

Given information:

Given figure

  

Calculation:

Consider the given parallelepiped having $u,v$ and $w$ as adjacent edges as shown in figure:

The geometric property of the cross product states that $\Vert u\times v\Vert$ is area of parallelogram that has $u$ and $v$ as adjacent sides.

Similarly $\Vert v\times w\Vert$ is area of parallelogram that has $v$ and $w$ as adjacent sides, and $\Vert w\times u\Vert$ is area of parallelogram that has $w$ and $u$ as adjacent sides.

The surface area $\left(S\right)$ of parallelepiped is sum of area of all 6 parallelograms faces, where area of parallel faces is same:

  $S=2\left(\Vert u\times v\Vert +\Vert v\times w\Vert +\Vert w\times u\Vert \right)$

Use the vector $u=\langle 4,-4,2\rangle ,v=\langle 0,4,-3\rangle$ and $w=\langle 1,3,4\rangle$ to find the surface $S$ . Thus you can see how adjacent edges can be used to find the surface area.