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Chapter 8: Problem 5

Sketch the graph of each equation in a three dimensional coordinate system. $$x+y-z=3$$

Short Answer

Expert verified
Plot the points (3, 0, 0), (0, 3, 0), and (0, 0, -3); connect them to form a plane.

Step by step solution

01

Understand the Equation

The given equation is a linear equation in three variables: \[ x + y - z = 3 \]This represents a plane in 3D space. To sketch it, we need to determine the points where this plane intersects the coordinate axes.
02

Find the Intercepts

To find the x-intercept, set both y and z to 0, \[ x + 0 - 0 = 3 \]\[ x = 3 \]Thus, the x-intercept is (3, 0, 0).To find the y-intercept, set both x and z to 0, \[ 0 + y - 0 = 3 \]\[ y = 3 \]Thus, the y-intercept is (0, 3, 0).To find the z-intercept, set both x and y to 0, \[ 0 + 0 - z = 3 \]\[ -z = 3 \rightarrow z = -3 \]Thus, the z-intercept is (0, 0, -3).
03

Plot the Intercepts

Mark the intercept points on the three-dimensional coordinate system: (3, 0, 0), (0, 3, 0), and (0, 0, -3). These points help in visualizing the plane on the graph.
04

Draw the Plane

Connect the intercept points to form a triangular section of the plane within the 3D coordinate system. Extend this triangular section to suggest the infinite plane.

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Key Concepts

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

Plane Equation
A plane in three-dimensional geometry can be understood through its equation. The general form of a plane equation is \(Ax + By + Cz = D\). Here, \(A\), \(B\), and \(C\) are constants that describe the orientation of the plane, and \(D\) is a constant that determines its position.
In our example, the plane equation is given by \(x + y - z = 3\). This means the plane is tilted in such a way that it crosses all three coordinate axes鈥攛, y, and z.
This tilting and crossing depend on the coefficients of x, y, and z. Because our given plane equation has all coefficients as 1 or -1, the plane is symmetrically inclined with respect to the axes.
Intercepts
Intercepts are crucial for sketching the graph of a plane. They show where the plane intersects each of the coordinate axes.
  • To find the x-intercept, set y and z to 0 in the plane equation \(x + y - z = 3\). This simplifies to \(x = 3\), giving the x-intercept as (3, 0, 0).
  • For the y-intercept, set x and z to 0. This simplifies to \(y = 3\), giving the y-intercept as (0, 3, 0).
  • For the z-intercept, set x and y to 0. This simplifies to \(\-z = 3\), which results in \(z = -3\), giving the z-intercept as (0, 0, -3).
These intercepts provide specific points through which the plane passes, making it easier to visualize and sketch the plane in 3D geometry.
Three-Dimensional Geometry
Three-dimensional geometry extends beyond the x and y axes. It includes an additional axis, the z-axis, allowing a more comprehensive view of shapes and planes.
In a 3D coordinate system, every point is defined by three coordinates (x, y, z). Understanding the relationships and intersections among these coordinates helps in visualizing more complex geometrical shapes.
  • The 3D coordinate system helps in presenting real-world problems where height (z-axis) is an essential aspect.
  • When graphing in 3D, it's important to project these points accurately to understand the object鈥檚 spatial arrangement.
  • Connecting the intercept points found for the plane equation \(x + y - z = 3\), (3, 0, 0), (0, 3, 0), and (0, 0, -3), illustrates the triangular facet of the plane in 3D space.
This process emphasizes the importance of the intercepts and how they guide the spatial arrangement of the plane.

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