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Chapter 6: Problem 50

What is a polar equation?

Short Answer

Expert verified
A polar equation is a mathematical equation expressed in polar coordinates \(r\) and \(\(\theta\)\), which represent the distance from the origin and the angle from the positive x-axis respectively.

Step by step solution

01

Definition of a Polar Equation

A polar equation is a mathematical relationship expressed in terms of polar coordinates \(r\) and \(\(\theta\)\), where \(r\) is the distance from the origin (pole) and \(\theta\) is the angle from the positive half of x-axis (polar axis).
02

Examples of Polar Equations

A simple polar equation could be \(r = a\), which would represent a circle with radius \(a\) centered at the origin. Another example might be \(\theta = a\), which would represent a line with angle \(a\) from the x-axis.
03

Polar vs Cartesian Equations

It's useful to understand that polar equations can often be transformed into Cartesian equations (using x and y) by using the relationships \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\). Similarly, a Cartesian equation can be transformed into a polar equation using \(r = \sqrt{x^2 + y^2}\) and \(\theta = \tan^{-1}(y/x)\).

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Most popular questions from this chapter

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 \(\mathbf{w}.\) $$\mathbf{v}=2 \mathbf{i}+4 \mathbf{j}, \quad \mathbf{w}=-3 \mathbf{i}+6 \mathbf{j}$$

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

Explain how to find the dot product of two vectors.

Find the angle, in degrees, between \(\mathbf{v}\) and \(\mathbf{w}.\) $$\mathbf{v}=3 \cos \frac{5 \pi}{3} \mathbf{i}+3 \sin \frac{5 \pi}{3} \mathbf{j}, \quad \mathbf{w}=2 \cos \pi \mathbf{i}+2 \sin \pi \mathbf{j}$$

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