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Unit vectors in polar coordinates Vectors in R may also be
Chapter 14, Problem 52AE(choose chapter or problem)
Unit vectors in polar coordinates
Vectors in \(\mathbf{R}^{2}\) may also be expressed in terms of polar coordinates. The standard coordinate unit vectors in polar coordinates are denoted \(\mathbf{u}_{r} \text { and } \mathbf{u}_{\theta}\) (see Figure). Unlike the coordinate unit vectors in Cartesian coordinates, \(\mathbf{u}_{r} \text { and } \mathbf{u}_{\theta}\) change their direction depending on the point \((r, \theta)\). Use the figure to show that for r > 0. The following relationships between the unit vectors in Cartesian and polar coordinates hold:
\(\mathbf{u}_{r}=\cos \theta \mathbf{i}+\sin \theta \mathbf{j}\) \(\mathbf{i}=\mathbf{u}_{r} \cos \theta-\mathbf{u}_{\theta} \sin \theta\)
\(\mathbf{u}_{\theta}=-\sin \theta \mathbf{i}+\cos \theta \mathbf{j}\) \(\mathbf{j}=\mathbf{u}_{r} \sin \theta+\mathbf{u}_{\theta} \cos \theta\)
Questions & Answers
QUESTION:
Unit vectors in polar coordinates
Vectors in \(\mathbf{R}^{2}\) may also be expressed in terms of polar coordinates. The standard coordinate unit vectors in polar coordinates are denoted \(\mathbf{u}_{r} \text { and } \mathbf{u}_{\theta}\) (see Figure). Unlike the coordinate unit vectors in Cartesian coordinates, \(\mathbf{u}_{r} \text { and } \mathbf{u}_{\theta}\) change their direction depending on the point \((r, \theta)\). Use the figure to show that for r > 0. The following relationships between the unit vectors in Cartesian and polar coordinates hold:
\(\mathbf{u}_{r}=\cos \theta \mathbf{i}+\sin \theta \mathbf{j}\) \(\mathbf{i}=\mathbf{u}_{r} \cos \theta-\mathbf{u}_{\theta} \sin \theta\)
\(\mathbf{u}_{\theta}=-\sin \theta \mathbf{i}+\cos \theta \mathbf{j}\) \(\mathbf{j}=\mathbf{u}_{r} \sin \theta+\mathbf{u}_{\theta} \cos \theta\)
ANSWER:
Solution 52AEPlaner vectors can be represented using Cartesian coordinate or polar coordinates in 2D plane. The relationship between unit vectors of Cartesian and polar coordinate is given below.Consider the figure gi