2.4 The cross product  (Page 6/16)

Note that, as the name indicates, the triple scalar product produces a scalar. The volume formula just presented uses the absolute value of a scalar quantity.

Proof

The area of the base of the parallelepiped is given by $\Vert \text{v}\times \text{w}\Vert .$ The height of the figure is given by $\Vert {\text{proj}}_{\text{v×w}}\text{u}\Vert .$ The volume of the parallelepiped is the product of the height and the area of the base, so we have

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Calculating the volume of a parallelepiped

Let $\text{u}=⟨\mathrm{-1},\mathrm{-2},1⟩,\text{v}=⟨4,3,2⟩,\text{and}\text{w}=⟨0,\mathrm{-5},\mathrm{-2}⟩.$ Find the volume of the parallelepiped with adjacent edges $\text{u},\text{v},\text{and}\text{w}$ ( [link] ).

We have

$\begin{array}{cc}\hfill \text{u}·\left(\text{v}\times \text{w}\right)& =\left|\begin{array}{ccc}\hfill \mathrm{-1}& \hfill \mathrm{-2}& \hfill 1\\ \hfill 4& \hfill 3& \hfill 2\\ \hfill 0& \hfill \mathrm{-5}& \hfill \mathrm{-2}\end{array}\right|=\left(\mathrm{-1}\right)\left|\begin{array}{cc}\hfill 3& \hfill 2\\ \hfill \mathrm{-5}& \hfill \mathrm{-2}\end{array}\right|+2\left|\begin{array}{cc}\hfill 4& \hfill 2\\ \hfill 0& \hfill \mathrm{-2}\end{array}\right|+\left|\begin{array}{cc}\hfill 4& \hfill 3\\ \hfill 0& \hfill \mathrm{-5}\end{array}\right|\hfill \\ & =\left(\mathrm{-1}\right)\left(\mathrm{-6}+10\right)+2\left(\mathrm{-8}-0\right)+\left(\mathrm{-20}-0\right)\hfill \\ & =\mathrm{-4}-16-20\hfill \\ & =\mathrm{-40.}\hfill \end{array}$

Thus, the volume of the parallelepiped is $\left|\mathrm{-40}\right|=40$ units ³ .

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Applications of the cross product

The cross product appears in many practical applications in mathematics, physics, and engineering. Let’s examine some of these applications here, including the idea of torque, with which we began this section. Other applications show up in later chapters, particularly in our study of vector fields such as gravitational and electromagnetic fields ( Introduction to Vector Calculus ).

We have seen how to use the triple scalar product and how to find a vector orthogonal to a plane. Now we apply the cross product to real-world situations.

Sometimes a force causes an object to rotate. For example, turning a screwdriver or a wrench creates this kind of rotational effect, called torque.