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Section 1.11 Polar basis vectors

Curvilinear basis vectors are vector fields, that is, they depend on the point where they are located. So the relationship between two such bases also depends on location.

Activity 1.11.1.

At the point \(P\) shown below, first draw the rectangular basis vectors \(\xhat\) and \(\yhat\text{,}\) then draw the polar basis vectors \(\rhat\) and \(\phat\text{.}\) Finally, use your drawing to work out the relationship between these basis vectors. Make sure that your formula works for every quadrant. Note: You can move the point \(P\) in the figures below.

Figure 1.11.1. Polar Coordinates.
Hint
Figure 1.11.2. Draw the polar basis vectors at the point \(P\text{.}\)