Problem 3 - Calculus for Biology and Medicine

Question

Locate the following points in a three-dimensional Cartesian coordinate system: (a) \((1,3,2)\) (b) \((-1,-2,1)\) (c) \((0,1,2)\) (d) \((2,0,3)\)

Step-by-Step Solution

Answer

Plot each point accordingly in 3D space using its coordinates on the x, y, and z axes.

Step 1: Understanding the Coordinate System

A three-dimensional Cartesian coordinate system consists of three axes: the x-axis, y-axis, and z-axis. Each point is represented by an ordered triplet \(x, y, z\), where \(x\) is the position on the x-axis, \(y\) is the position on the y-axis, and \(z\) is the position on the z-axis.

Step 2: Plotting Point (1, 3, 2)

To locate point \(1, 3, 2\), start at the origin \(0, 0, 0\). Move 1 unit along the x-axis to the right. From there, move 3 units in the positive direction of the y-axis. Finally, move 2 units up in the positive direction of the z-axis. Mark this location as point \(1, 3, 2\).

Step 3: Plotting Point (-1, -2, 1)

To locate point \(-1, -2, 1\), start at the origin \(0, 0, 0\). Move 1 unit along the x-axis to the left (since -1 is negative). Then, move 2 units in the negative direction of the y-axis. Finally, move 1 unit up in the positive direction of the z-axis. Mark this location as point \(-1, -2, 1\).

Step 4: Plotting Point (0, 1, 2)

To locate point \(0, 1, 2\), start at the origin \(0, 0, 0\). Do not move along the x-axis since the x-coordinate is 0. Move 1 unit in the positive direction of the y-axis. Then, move 2 units up in the positive direction of the z-axis. Mark this location as point \(0, 1, 2\).

Step 5: Plotting Point (2, 0, 3)

To locate point \(2, 0, 3\), start at the origin \(0, 0, 0\). Move 2 units along the x-axis to the right. Do not move along the y-axis since the y-coordinate is 0. Finally, move 3 units up in the positive direction of the z-axis. Mark this location as point \(2, 0, 3\).

Key Concepts

Understanding Three-Dimensional Coordinates

In the three-dimensional Cartesian coordinate system, each point is identified by three numbers, usually written as \((x, y, z)\). These numbers indicate the distances from a point to three mutually perpendicular planes that meet at a point called the origin.

• X-coordinate: This number tells us how far to move along the x-axis, which is typically left or right in diagrams.
• Y-coordinate: This number represents the movement along the y-axis, typically forward or backward.
• Z-coordinate: This one tells us how far to move up or down along the z-axis.

Each of these directions is perpendicular to the others, forming a 3D space. For instance, the point \((1, 3, 2)\) indicates moving 1 unit along the x-axis, 3 units along the y-axis, and 2 units along the z-axis.

Plotting Points in 3D Space

Plotting points in three-dimensional space involves moving in three directions: horizontally, vertically, and depth-wise. Let's break it down using our exercises:- **Moving from the Origin**: For each point, always start from the origin \((0, 0, 0)\).- **Moving Horizontally and Vertically**: Your first movement is along the x-axis, followed by the y-axis. Be cautious of negative coordinates, which mean moving in the opposite direction.- **Adding Depth (the Z-axis)**: Finally, consider the z-coordinate to move either up or down.For example, to locate \((-1, -2, 1)\), from the origin move 1 unit left on the x-axis, then 2 units backward on the y-axis, and finally 1 unit upward on the z-axis. The main challenge in 3D plotting is visualizing these movements in a physical space which requires imagining the z-axis popping out of the paper or screen.

Identifying the Coordinate Axes

The coordinate axes in a three-dimensional Cartesian system are similar to those in two dimensions, but with an extra axis added for depth. Understanding these axes is key to effectively navigating 3D space:

• X-Axis: This axis can be visualized as running left and right, helping dictate the horizontal placement of points.
• Y-Axis: Typically thought of as running forward and backward, it defines the vertical placement of points in many 2D planes.
• Z-Axis: Unlike the others, the z-axis stands vertically, like a pole that moves up and down, adding depth to our perception of space.

When dealing with coordinate axes, remember that each axis is perpendicular to the others, creating a comprehensive 3D grid. This is essential for mapping out intricate points as in your exercises, where accuracy is determined by correctly using each axis.