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Chapter 6: Problem 8

Draw a set of \(x-, y-\), and \(z\) -axes and plot the following points: a. \(A(1,0,0)\) c. \(C(0,0,-3)\) e. \(E(2,0,3)\) b. \(B(0,-2,0)\) d. \(D(2,3,0)\) f. \(F(0,2,3)\)

Short Answer

Expert verified
Plot points in 3D space on the x-y-z axes using their coordinates.

Step by step solution

01

Understand the Axes

The x-, y-, and z-axes are part of a 3D coordinate system. The x-axis is usually horizontal and runs left to right, the y-axis is vertical and runs up and down, and the z-axis runs perpendicular to both, coming out of or going into the page or screen.
02

Plot Point A

Point A has coordinates \((1,0,0)\). This means move 1 unit along the x-axis, and do not move along the y- or z-axis. This point will be on the x-axis.
03

Plot Point B

Point B has coordinates \((0,-2,0)\). This means do not move along the x-axis, move 2 units down on the y-axis. This point lies on the y-axis in the negative direction.
04

Plot Point C

Point C has coordinates \((0,0,-3)\). This means do not move along the x- or y-axis, and move 3 units in the negative direction along the z-axis.
05

Plot Point D

Point D is \((2,3,0)\). Move 2 units along the x-axis and 3 units up the y-axis. This point lies on the x-y plane.
06

Plot Point E

Point E is \((2,0,3)\). Move 2 units along the x-axis, do not move along the y-axis, and move 3 units in the positive direction along the z-axis.
07

Plot Point F

Point F has coordinates \((0,2,3)\). This means you do not move along the x-axis, move 2 units up the y-axis, and then move 3 units in the positive z-axis direction.
08

Review the Plot

With all points plotted, ensure they align with their respective coordinates. Double-check that each point adheres to its corresponding plane as needed.

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Key Concepts

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

Plotting Points
Plotting points in a 3-dimensional coordinate system can initially seem a bit challenging, but it's all about understanding the role each coordinate plays. A standard point in 3D space is represented by three coordinates:
  • The first coordinate, recognized as the x-coordinate, tells you how far to move along the x-axis.
  • The second, or y-coordinate, describes the movement along the y-axis.
  • The third coordinate, the z-coordinate, indicates your movement along the z-axis.
To plot a point such as \((1, 0, 0)\), you move 1 unit on the x-axis, with no movement along the y and z axes. Hence, understanding these coordinates allows us to correctly locate points in 3D space.
x-y-z Axes
The x-, y-, and z-axes form the foundation of the 3D coordinate system. Think of them as the number lines placed in 3D space:
  • The x-axis runs horizontally, left to right.
  • The y-axis runs vertically, up and down.
  • The z-axis adds depth, coming out of or going into the page or screen.
These axes intersect at the origin, which is \((0, 0, 0)\). They help us navigate through 3D space, essentially serving as directions or paths to follow when plotting any point. Precisely knowing your position along each axis is crucial for properly plotting points.
Coordinate Planes
In a 3D coordinate system, planes are flat, two-dimensional surfaces extended infinitely. The coordinate planes are formed by the axes:
  • The x-y plane is crafted by the intersection of the x- and y- axes, ignoring z coordinates. Points like \((2, 3, 0)\) lie on this plane.
  • The x-z plane arises from the x- and z-axes, where y is zero.
  • The y-z plane is derived from the y- and z-axes, with x-coordinates set at zero.
These planes serve as critical reference points, simplifying the process of understanding complex 3D spaces. By focusing on the coordinates relevant to each plane, plotting becomes straightforward.

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Most popular questions from this chapter

a. Draw a set of \(x-, y-\), and \(z\) -axes and plot the following points: \(A(3,2,-4)\) \(B(1,1,-4),\) and \(C(0,1,-4)\) b. Determine the equation of the plane containing the points \(A, B,\) and \(C .\)

In \(\triangle A B C,\) a median is drawn from vertex \(A\) to the midpoint of \(B C\) which is labelled \(D .\) If \(\overrightarrow{A B}=\vec{b}\) and \(\overrightarrow{A C}=\vec{c},\) prove that \(\overrightarrow{A D}=\frac{1}{2} \vec{b}+\frac{1}{2} \vec{c}\).

Draw two vectors, \(\vec{a}\) and \(\vec{b}\), that do not have the same magnitude and are noncollinear. Using the vectors you drew, construct the following: a. \(2 \vec{a}\) b. \(3 \vec{b}\) c. \(-3 \vec{b}\) d. \(2 \vec{a}+3 \vec{b}\) e. \(2 \vec{a}-3 \vec{b}\)

For \(A(-1,3)\) and \(B(2,5),\) draw a coordinate plane and place the points on the graph. a. Draw vectors \(\overrightarrow{A B}\) and \(\overrightarrow{B A}\), and give vectors in component form equivalent to each of these vectors. b. Determine \(|\overrightarrow{O A}|\) and \(|\overrightarrow{O B}|\) c. Calculate \(|\overrightarrow{A B}|\) and state the value of \(|\overrightarrow{B A}|\)

If \(*\) is an operation on a set, \(S\), the element \(x\), such that \(a^{*} x=a\), is called the identity element for the operation *. a. For the addition of numbers, what is the identity element? b. For the multiplication of numbers, what is the identity element? c. For the addition of vectors, what is the identity element? d. For scalar multiplication, what is the identity element?
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