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Section 1.20 Linearity of the Dot and Cross Products

The linearity of the dot and cross products follows immediately from their algebraic definitions. However, the above derivations of the algebraic formulas from the geometric definitions assumed without comment that both the dot and cross products distribute over addition. To complete the derivation, we must check that linearity follows from the geometric definition. These geometric derivations are shown in Figure 1.20.1 and Figure 1.20.2 below.

For the dot product, we must show that

\begin{gather*} (\vv+\uu) \cdot \ww = \vv\cdot\ww + \uu\cdot\ww \end{gather*}

which is equivalent to showing that the projection of \(\vv+\uu\) along \(\ww\) is the sum of the projections of \(\vv\) and \(\uu\text{,}\) which is immediately obvious from Figure 1.20.1. In this figure, \(\vv\) is shown in blue, \(\uu\) in red, their sum in green, and \(\ww\) in black.

Figure 1.20.1. A geometric proof of the linearity of the dot product.

For the cross product, we must show that

\begin{gather} \ww \times (\vv+\uu) = \ww\times\vv + \ww\times\uu\label{crossgeom}\tag{1.20.1} \end{gather}

which follows with a little thought from Figure 1.20.2, which uses the same color scheme as before.

Figure 1.20.2. A geometric proof of the linearity of the cross product.

Consider in turn the vectors \(\vv\text{,}\) \(\uu\text{,}\) and \(\vv+\uu\text{.}\) The cross product of each of these vectors with \(\ww\) is proportional to its projection perpendicular to \(\ww\text{.}\) These projections are shown as solid lines in the figure. Since the projections lie in the plane perpendicular to \(\ww\text{,}\) they can be combined into the triangle shown in the middle of the figure. Two of the vectors making up the sides of a triangle add up to the third; in this case, the sides are the projections of \(\vv\text{,}\) \(\uu\text{,}\) and \(\vv+\uu\text{,}\) and the latter is clearly the sum of the first two. But each cross product is now just a rotation of one of the sides of this triangle, rescaled by the length of \(\ww\text{;}\) these are the arrows perpendicular to the faces of the prism. Two of these vectors therefore still add to the third, as indicated by the vector triangle in front of the prism. This establishes (1.20.1).