To graph a cylindrical point in space, you need to understand how cylindrical coordinates $(r, \theta, z)$ relate to the standard 3D Cartesian coordinates $(x, y, z)$. Here's a step-by-step explanation of how to plot the point $(2, 3\pi/4, -5)$:

Components of Cylindrical Coordinates:

• $r = 2$: This is the radial distance from the origin (along the xy-plane). It's how far the point is from the z-axis.
• $\theta = 3\pi/4$: This is the angle measured counterclockwise from the positive x-axis. It determines the direction in the xy-plane.
• $z = -5$: This is the height of the point, indicating the position along the z-axis. A negative value means the point is below the xy-plane.

Steps to Graph the Point:

1. Start with the xy-plane (base):

   • Imagine you're looking at the xy-plane, which is flat.
   • The radius $r = 2$ tells you the distance from the origin along the xy-plane. So, from the origin, move 2 units outward from the z-axis.

2. Determine the direction using $\theta = 3\pi/4$:

   • The angle $3\pi/4$ radians is measured counterclockwise from the positive x-axis.
   • $3\pi/4$ radians is equivalent to 135°. This places you in the second quadrant of the xy-plane. From the origin, rotate 135° from the positive x-axis.

3. Mark the point in the xy-plane:

   • After rotating 135°, plot the point 2 units away from the origin in that direction (along the ray from the origin).

4. Move along the z-axis:

   • Now, consider the z-coordinate $z = -5$. This means the point is 5 units below the xy-plane.
   • From the point you've marked in the xy-plane, move straight downward (parallel to the z-axis) by 5 units.

Identifying the Octant:

To determine the octant, consider the following:

• $r = 2$ gives us the distance from the origin but not the sign of the coordinates. This depends on $\theta$.
• $\theta = 3\pi/4$ corresponds to the second quadrant of the xy-plane, where $x$ is negative, and $y$ is positive.
• $z = -5$ means the point is below the xy-plane.

In Cartesian coordinates:

• The angle $3\pi/4$ tells us that $x < 0$ and $y > 0$, while $z < 0$.

Therefore, the point lies in the fifth octant, where $x$, $z$ are negative, and $y$ is positive.

Summary:

• $r = 2$: Move 2 units from the origin in the direction given by $\theta$.
• $\theta = 3\pi/4$: Rotate 135° counterclockwise from the positive x-axis.
• $z = -5$: Move 5 units downward from the xy-plane.
• The point lies in the fifth octant.

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Would you like more details on any of the steps? Here are five related questions:

1. How do cylindrical coordinates relate to spherical coordinates?
2. What is the formula to convert cylindrical coordinates to Cartesian coordinates?
3. How would the graph of this point change if the $z$-coordinate were positive?
4. Can you describe the process of plotting a cylindrical point in 2D polar coordinates?
5. How do the octants of 3D space differ from the quadrants in 2D space?

Tip: Always keep track of the signs of the coordinates when identifying the octant!