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Chapter 6: Problem 105

Determine whether each statement makes sense or does not make sense, and explain your reasoning. After plotting the point with rectangular coordinates \((0,-4),\) I found polar coordinates without having to show any work.

Short Answer

Expert verified
The statement does not make sense because converting from rectangular to polar coordinates requires the use of formulas and some work to calculate the values.

Step by step solution

01

Understand the Rectangular Coordinates

The given rectangular coordinates are (0,-4). This indicates that the point is 4 units below the origin on the y-axis.
02

Convert Rectangular to Polar Coordinates

To convert rectangular coordinates to polar, use the formula \[r = \sqrt{x^2 + y^2}\] for the radial coordinate, and \[\theta = tan^{-1}(y/x)\] for the angular coordinate. However, as our x component is zero, the formula for 胃 is not defined, instead 胃 will be either \(\pi/2\) or \(-\pi/2\) (90 or -90 degrees) depending on the y component. In this case, since y is negative, 胃 is -90 degrees. The value of r can be found to be 4, despite y being negative, as r is a distance and hence always positive.
03

Verify the Statement

After conversion, the polar coordinates of the point are (4, -90 degrees). As this was not a trivial conversion, the statement 'I found polar coordinates without having to show any work' does not make sense because work was indeed required for the conversion.

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

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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 \(\mathbf{w}.\) $$\mathbf{v}=\mathbf{i}+3 \mathbf{j}, \quad \mathbf{w}=-2 \mathbf{i}+5 \mathbf{j}$$

Let $$\mathbf{u}=-\mathbf{i}+\mathbf{j}, \quad \mathbf{v}=3 \mathbf{i}-2 \mathbf{j}, \quad \text { and } \quad \mathbf{w}=-5 \mathbf{j}$$ Find each specified scalar or vector. $$4 \mathbf{u} \cdot(5 \mathbf{v}-3 \mathbf{w})$$
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