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Section 8.3 Visualizing the Dot Product in Higher Dimensions

The figure below shows a representation of the dot product in five dimensions, similar to the one given in Section 1.8. However, unlike that two-dimensional representation, we can not display the five (vector) components as orthogonal arrows. Instead, the sliders control the values of the (scalar) components, with the five components of \(\vv\) shown on the left in brown, and the five components of \(\ww\) shown on the right in blue. As in Figure 1.12, the historgram boxes provide a visual representation of the selected components, with corresponding products shown in green.

Figure 8.3. A representation of the dot product in five dimensions.

The same technique can be used to represent the dot product in any (relatively small) number of dimensions. This representation is based on the algebraic formula (1.7.5) for the dot product, and assumes that the vectors are expressed in terms of an orthonormal basis.