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Chapter 6: Problem 63

Convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation. $$r \sin \theta=3$$

Short Answer

Expert verified
The rectangular equivalent of the given polar equation \(r \sin \theta = 3\) is \(y=3\). The graph of this equation is a straight horizontal line passing through \(y=3\).

Step by step solution

01

Conversion of Polar to Rectangular equation

The given polar equation is \(r \sin \theta = 3\). In a rectangular coordinate system, \(r \sin \theta\) corresponds to the coordinate \(y\) (since \(r \sin \theta\) gives the height or 'y' value in the Cartesian system). So, by replacing \(r \sin \theta\) with \(y\), we get \(y = 3\).
02

Graphing the Rectangular equation

To graph the equation \(y = 3\) on a rectangular coordinate system, plot a straight horizontal line that intersects the y-axis at \(y = 3\). The line will extend to infinity on both left and right.

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Most popular questions from this chapter

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. $$\mathbf{u} \cdot \mathbf{v}=\mathbf{v} \cdot \mathbf{u}$$

Explain how to find the dot product of two vectors.

Graph \(r_{1}\) and \(r_{2}\) in the same polar coordinate system. What is the relationship between the two graphs? $$r_{1}=4 \cos 2 \theta, r_{2}=4 \cos 2\left(\theta-\frac{\pi}{4}\right)$$

What are orthogonal vectors?

Determine whether v and w are parallel, orthogonal, or neither. $$\mathbf{v}=3 \mathbf{i}-5 \mathbf{j}, \quad \mathbf{w}=6 \mathbf{i}-10 \mathbf{j}$$
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