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Chapter 6: Problem 55

Describe the test for symmetry with respect to the pole.

Short Answer

Expert verified
The test for symmetry with respect to the pole in a polar equation implies replacing \( r \) with \( -r \) and \( 胃 \) with \( -胃 \). The equation is symmetric with respect to the pole if it remains unchanged.

Step by step solution

01

Understand Polar Coordinates

In polar coordinates, a point in the plane is described by its distance r from the origin and the angle 胃 it forms with the positive x-axis. The polar coordinates (r, 胃) corresponds to the point (r cos 胃, r sin 胃) in Cartesian coordinates.
02

Concept of Symmetry in Polar Equations

A polar equation is said to be symmetric with respect to the origin if the equation possesses the same characteristics throughout, regardless of the angle. Essentially, we can go around the pole 360掳 and the graph looks the same all the way round.
03

Test for Symmetry with respect to the Pole

The test for symmetry with respect to the pole, that is the origin, consists of switching \( r \) for \( -r \) and \( 胃 \) for \( -胃 \) in the given polar equation. If the equation remains unchanged, then it is symmetric with respect to the pole. Mathematically, a polar equation \( r = f(胃) \) is symmetric with respect to the origin if and only if \( r = f(-胃) \) and \( r = -f(胃) \).

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Most popular questions from this chapter

A force is given by the vector \(\mathbf{F}=3 \mathbf{i}+2 \mathbf{j} .\) The force moves an object along a straight line from the point (4,9) to the point \((10,20) .\) Find the work done if the distance is measured in feet and the force is measured in pounds.

Find \(\text {pro}_{\mathbf{w}} \mathbf{V}\) Then decompose v into two vectors, \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2},\) where \(\mathbf{v}_{1}\) is parallel to \(\mathbf{w}\) and \(\mathbf{v}_{2}\) is orthogonal to \(\mathbf{w}.\) $$\mathbf{v}=\mathbf{i}+2 \mathbf{j}, \quad \mathbf{w}=3 \mathbf{i}+6 \mathbf{j}$$

Use the dot product to determine whether v and w are orthogonal. $$\mathbf{v}=3 \mathbf{i}, \quad \mathbf{w}=-4 \mathbf{i}$$

Find the angle between \(\mathbf{v}\) and \(\mathbf{w} .\) Round to the nearest tenth of a degree. $$\mathbf{v}=-3 \mathbf{i}+2 \mathbf{j}, \quad \mathbf{w}=4 \mathbf{i}-\mathbf{j}$$

Use the vectors $$\mathbf{u}=a_{1} \mathbf{i}+b_{1} \mathbf{j}, \quad \mathbf{v}=a_{2} \mathbf{i}+b_{2} \mathbf{j}, \quad \text { and } \quad \mathbf{w}=a_{3} \mathbf{i}+b_{3} \mathbf{j},$$ to prove the given property. $$\mathbf{u} \cdot \mathbf{v}=\mathbf{v} \cdot \mathbf{u}$$
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