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Magazine Chapter 7: Problem 5
Graph the given point. $$ (6,-2,0) $$
Short Answer
Expert verified
Plot the point (6,-2,0) on a 3D graph along the x and y axes, keeping z constant at 0.
Step by step solution
01
Understand the Point's Coordinates
The given point is \((6, -2, 0)\). It is represented in a three-dimensional coordinate system as \((x, y, z)\).This means:- The \(x\)-coordinate is 6,- The \(y\)-coordinate is -2,- The \(z\)-coordinate is 0.
02
Set Up the Three-Dimensional Axes
Visualize or draw the three-dimensional coordinate system. The system includes:- An \(x\)-axis which typically runs horizontally,- A \(y\)-axis which often runs vertically in two dimensions and diagonally in three dimensions, and - A \(z\)-axis which represents the depth or height, running perpendicular to the \(x\) and \(y\) axes.
03
Locate the \(x\)-Coordinate
Move to the right on the \(x\)-axis 6 units from the origin, \((0,0,0)\). This is because the \(x\)-coordinate is 6.
04
Locate the \(y\)-Coordinate
From the point \((6,0,0)\), move 2 units downward along the \(y\)-axis (or towards the negative direction since it's -2). This gives the new position \((6,-2,0)\).
05
Verify the \(z\)-Coordinate
Check that the \(z\)-coordinate is 0. This means there is no movement along the \(z\)-axis (no change in depth), so \((6, -2, 0)\) remains on the plane created by the \(x\) and \(y\) axes.
06
Plot and Review
Now you can plot the point \((6, -2, 0)\) on your three-dimensional graph and verify that every movement corresponds to the respective coordinate. The point should lie on the \(xy\)-plane since the \(z\)-coordinate is 0.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Graphing Points
Graphing points is the process of plotting a location based on its coordinates in a specified plane. When given a point like \((6, -2, 0)\), you are dealing with a three-dimensional coordinate system:
- First, identify the coordinates: \(x = 6\), \(y = -2\), \(z = 0\).
- This point signifies its position in a 3D space using values for each axis.
- Visualize the placement of this point by moving along the axes from the origin \((0, 0, 0)\).
Three-Dimensional Graphs
Three-dimensional graphs expand upon the two-dimensional plane by adding depth, offering a more comprehensive view of space. In a 3D graph:
- The \(x\)-axis runs horizontally, similar to its role in 2D graphs.
- The \(y\)-axis departs from verticality, leaning towards a diagonal position often.
- The \(z\)-axis introduces a new dimension, shooting perpendicularly to create depth.
Coordinate Geometry
Coordinate geometry, sometimes called analytic geometry, uses a coordinate system to explore geometric figures. With the addition of the third dimension, this branch of mathematics enables understanding of shapes and spaces:
- Defines points by their position relative to three axes, \(x\), \(y\), and \(z\).
- Illustrates concepts like distance, slope, and planes using numeric coordinates.
- Facilitates the exploration and visualization of spatial relationships.
