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Chapter 3: Problem 6

Graph each point in coordinate space. $$ (0,0,4) $$

Short Answer

Expert verified
The point (0,0,4) is located 4 units above the origin on the z-axis.

Step by step solution

01

Understanding 3D Coordinates

In a three-dimensional coordinate system, points are given as ( 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.
02

Setting up the Axes

Draw three mutually perpendicular lines to represent the x, y, and z axes. Typically, the z-axis is vertical, the x-axis is horizontal and extends to the right, and the y-axis is horizontal and extends into the page.
03

Plotting the Point

Since the point is (0,0,4), it means the point is located at the origin on the x-axis and y-axis but 4 units above the origin on the z-axis. Place a dot at the position where z equals 4, which should be directly above the origin.

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Key Concepts

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

Graphing Points in 3D
When first learning to graph points in a three-dimensional coordinate system, it's like adding an extra layer to what you may already know from 2D graphing. Imagine a room where the floor represents the x-axis and the y-axis. If you hold a balloon in this room, the string's length determines how high it is off the ground, similar to the z-axis in 3D space.

For graphing the point (0,0,4), envision placing the balloon right above the corner where the two walls and floor meet (which is the origin), then lift it up to the height equivalent to 4 units. It stays in place in the air, not touching the walls or the floor. That's your point in 3D space!
Three-Dimensional Coordinates
Understanding three-dimensional, or 3D, coordinates means grasping how to describe the position of a point in space using three values. These values correspond to the distances along the x-axis, y-axis, and z-axis, usually denoted as \(x, y, z\).

In the context of our example, the point (0,0,4) has its \(x\) and \(y\) coordinates as zero because it's aligned with the origin in the horizontal plane, while its \(z\) value is 4 because it's four units above that plane. Always start counting from the origin, which is where all three axes meet, and you can move positively or negatively along these axes to find any point in space.
Plotting Points in Coordinate Space
Plotting a point like (0,0,4) in coordinate space involves visualising where the point exists relative to the origin. Setting up your axes correctly is crucial. Make sure they're perpendicular to each other, resembling corner edges of a box.

Now, since both \(x\) and \(y\) are zero, these two dimensions don't move the point left-right or forward-backward from the origin. Instead, just focus on the \(z\)-axis. Move straight up from the origin to a spot floating above it at what would be called '4 units up'. This is where you plot the point, like a firefly glowing quietly in that exact position.

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Most popular questions from this chapter

Solve each system. $$ \left\\{\begin{array}{l}{5 z+4 y=4} \\ {3 x-2 y=0} \\ {x+3 z=-8}\end{array}\right. $$

For each function, \(y\) varies directly as \(x\) If \(y=9\) when \(x=15,\) find \(x\) when \(y=6\)

Evaluate each expression for \(a=3\) and \(b=-5\) \(2 a+b\)

Solve each system by elimination. Check your answers. $$ \left\\{\begin{aligned} x-y-2 z &=4 \\\\-x+2 y+z &=1 \\\\-x+y-3 z &=11 \end{aligned}\right. $$

Solve each system by substitution. Check your answers. $$ \left\\{\begin{aligned} x-4 y+z &=6 \\ 2 x+5 y-z &=7 \\ 2 x-y-z &=1 \end{aligned}\right. $$
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