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Chapter 10: Problem 14

Give the rectangular coordinates of the point. $$[2,0]$$

Short Answer

Expert verified
The rectangular coordinates of the given point are \([2, 0]\).

Step by step solution

01

Identify the x-coordinate

Identify the x-coordinate of the point. The x-coordinate is 2, which means the point is located 2 units to the right of the origin (where the x-axis and y-axis intersect).
02

Identify the y-coordinate

Identify the y-coordinate of the point. The y-coordinate is 0, which means the point lies on the x-axis itself.
03

Present the rectangular coordinates

Combine the x-coordinate and y-coordinate to represent the rectangular coordinates of the point. The rectangular coordinates of the given point are \([2, 0]\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the X-Coordinate
In the world of rectangular coordinates, every point on a plane is represented by an ordered pair e.g., \([x, y]\). The first number in this pair is called the x-coordinate. It tells you how far left or right a point is from the origin. The origin is like the home base on a graph, located where the x-axis and the y-axis intersect, at point (0,0).
The value of the x-coordinate determines the horizontal position:
  • A positive x-coordinate means the point is to the right of the origin.
  • A negative x-coordinate places the point to the left of the origin.
  • If the x-coordinate is zero, the point is exactly on the y-axis.
Understanding the x-coordinate helps in visualizing the horizontal location of a point on a Cartesian plane.
Exploring the Y-Coordinate
The second number in the ordered pair, the y-coordinate, denotes the point's vertical position relative to the origin. While the x-coordinate describes horizontal movement, the y-coordinate focuses on the vertical dimension.
  • If the y-coordinate is positive, the point is above the x-axis.
  • A negative y-coordinate places the point below the x-axis.
  • When the y-coordinate is zero, the point is on the x-axis.
For the point \([2,0]\), the y-coordinate is zero, which means it rests directly on the x-axis. Recognizing the y-coordinate's value is crucial for determining how high or low the point is relative to the x-axis.
Graphing Points with Rectangular Coordinates
Graphing points with rectangular coordinates combines both the x-coordinate and y-coordinate to pin down the exact location of a point on the plane. To plot a point like \([2,0]\), start at the origin:
  • First, move 2 units right on the x-axis since the x-coordinate is 2.
  • Then, since the y-coordinate is 0, you don't move up or down.

Once placed, the graph reveals the point's spatial context, showing how it relates to other points and axes. This method streamlines understanding and visualizing geometric concepts, making it easier to explore mathematical relationships and solve problems confidently.

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Most popular questions from this chapter

Use this method to find the point(s) of self-intersection of each of the following curves. $$x(t)=t^{3}-4 t, \quad y(t)=t^{3}-3 t^{2}+2 t \quad t \text { real. }$$

The figure shows the graph of a function \(f\) continuously differentiable from \(x=a\) to \(x=b\) together with a polynomial approximation. Show that the length of this polygonal approximation can be written as the following Riemann sum: $$\sqrt{\left.1+[f^{\prime}\left(x_{1}^{*}\right)\right]^{2}} \Delta x_{1}+\cdots+\sqrt{1+\left[f^{\prime}\left(x_{n}^{*}\right)\right]^{2}} \Delta x_{n}$$ As \(\|P\|=\max \Delta x,\) tends to \(0,\) such Ricmann sums tend to $$\int_{a}^{b} \sqrt{\left.1+[f^{\prime}(x)\right]^{2}} d x$$

(a) Let \(a>0 .\) Find the length of the path traced out by $$\begin{array}{l}x(\theta) \quad 3 a \cos \theta+a \cos ^{2} \theta \\\y(\theta)=3 a \sin \theta-a \sin 3 \theta\end{array}$$ as \(\theta\) ranges from 0 to \(2 \pi\). (b) Show that this path can also be parametrized by $$x(\theta)=4 a \cos ^{3} \theta, \quad y(\theta)=4 a \sin ^{3} \theta \quad 0 \leq \theta \cdot 2 \pi$$

Find the a:ea of the surface generated by revolving the curve about the \(x\) -axis. \(f(x)=\sqrt{x} . \quad x \in[1,2]\).

Sketch the polar curve $$r=1-\cos \theta, \quad 0 \leq \theta \leq \pi$$ and calculate the length of the curve.
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