Question
Sketch the graph of x^2+z^2=1 in 3D space
Answers
GPT-4o mini
The equation \(x^2 + z^2 = 1\) represents a cylinder in 3D space. To understand this, let's break down the components of the equation:
1. **Types of Coordinates**: In three-dimensional space, we typically use (x, y, z) coordinates. In this equation, we notice that only \(x\) and \(z\) are included. This indicates that the shape is invariant with respect to the \(y\)-coordinate.
2. **Shape in X-Z Plane**: The equation \(x^2 + z^2 = 1\) describes a circle with a radius of 1 in the X-Z plane. The circle is centered at the origin (0, 0) of the X-Z plane.
3. **Extension in Y-Direction**: Since there's no \(y\) term in the equation, the cylinder extends infinitely in the positive and negative \(y\) directions. This means for every point on the circle defined by \(x^2 + z^2 = 1\), there is a corresponding line parallel to the \(y\)-axis extending indefinitely.
### Sketching the Graph:
To visualize this:
1. **Draw the Circle**: In the X-Z plane (the plane formed by the x-axis and z-axis), sketch a circle with a radius of 1 centered at the origin (0,0).
2. **Extend in Y-Direction**: From every point on the circumference of this circle, draw vertical lines extending both up and down along the \(y\)-axis. These lines represent the height of the cylinder at that x-z coordinate.
3. **Shape Overview**: The overall shape should look like a vertical cylinder (with a radius of 1) that runs indefinitely along the \(y\)-axis.
### Important Points:
- The cylinder has a circular cross-section of radius 1.
- It is symmetrical around the y-axis and extends infinitely in both directions along the y-axis.
This visual representation provides a clear understanding of how the 3D graph of the equation \(x^2 + z^2 = 1\) behaves.
1. **Types of Coordinates**: In three-dimensional space, we typically use (x, y, z) coordinates. In this equation, we notice that only \(x\) and \(z\) are included. This indicates that the shape is invariant with respect to the \(y\)-coordinate.
2. **Shape in X-Z Plane**: The equation \(x^2 + z^2 = 1\) describes a circle with a radius of 1 in the X-Z plane. The circle is centered at the origin (0, 0) of the X-Z plane.
3. **Extension in Y-Direction**: Since there's no \(y\) term in the equation, the cylinder extends infinitely in the positive and negative \(y\) directions. This means for every point on the circle defined by \(x^2 + z^2 = 1\), there is a corresponding line parallel to the \(y\)-axis extending indefinitely.
### Sketching the Graph:
To visualize this:
1. **Draw the Circle**: In the X-Z plane (the plane formed by the x-axis and z-axis), sketch a circle with a radius of 1 centered at the origin (0,0).
2. **Extend in Y-Direction**: From every point on the circumference of this circle, draw vertical lines extending both up and down along the \(y\)-axis. These lines represent the height of the cylinder at that x-z coordinate.
3. **Shape Overview**: The overall shape should look like a vertical cylinder (with a radius of 1) that runs indefinitely along the \(y\)-axis.
### Important Points:
- The cylinder has a circular cross-section of radius 1.
- It is symmetrical around the y-axis and extends infinitely in both directions along the y-axis.
This visual representation provides a clear understanding of how the 3D graph of the equation \(x^2 + z^2 = 1\) behaves.