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Chapter 29: Problem 7

In Exercises \(5-26,\) sketch the graphs of the given equations in the rectangular coordinate system in three dimensions. $$4 x-2 y+z-8=0$$

Short Answer

Expert verified
The plane intersects the x, y, and z axes at (2, 0, 0), (0, -4, 0), and (0, 0, 8), respectively.

Step by step solution

01

Understand the Equation Form

The given equation is a linear equation in three variables, representing a plane. It is written in the form: \( ax + by + cz + d = 0 \), where \( a = 4 \), \( b = -2 \), \( c = 1 \), and \( d = -8 \).
02

Find the X-intercept

To find the x-intercept, we set \( y = 0 \) and \( z = 0 \) in the equation and solve for \( x \). This gives us \( 4x - 8 = 0 \), thus \( x = 2 \). The x-intercept is \( (2, 0, 0) \).
03

Find the Y-intercept

To find the y-intercept, set \( x = 0 \) and \( z = 0 \) in the equation and solve for \( y \). This results in \( -2y - 8 = 0 \), so \( y = -4 \). The y-intercept is \( (0, -4, 0) \).
04

Find the Z-intercept

To find the z-intercept, set \( x = 0 \) and \( y = 0 \) in the equation and solve for \( z \). The equation simplifies to \( z - 8 = 0 \), thus \( z = 8 \). The z-intercept is \( (0, 0, 8) \).
05

Sketch the Plane Using Intercepts

Plot the intercepts found: \((2, 0, 0)\), \((0, -4, 0)\), and \((0, 0, 8)\) in a three-dimensional coordinate system. Draw lines connecting these intercepts to sketch the plane. In a 3D plot, these points form a triangle which is part of the plane.

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Key Concepts

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

Linear equations in 3D
Linear equations in a three-dimensional coordinate system express the relationship between three variables. In general, these equations are written in the form \( ax + by + cz + d = 0 \). When written like this, each component after the coefficients (\(a, b, c\)) represents a distinct variable (\(x, y, z\)).
These equations represent planes rather than lines, as they would in two dimensions. This means they describe a flat surface extending infinitely in all directions within the 3D space.
  • The coefficient \(a\) dictates the influence of the \(x\) variable on the plane.
  • Similarly, \(b\) and \(c\) affect the influence of \(y\) and \(z\) variables respectively.
  • The constant \(d\) determines the plane's position relative to the origin of the coordinate system.
Understanding linear equations in three dimensions is foundational for sketching and analyzing planes, crucial activities in various fields such as engineering, physics, and computer graphics.
Intercepts in 3D
Intercepts in a 3D coordinate system are where a plane meets the coordinate axes. Finding intercepts for a 3D equation follows a similar process to that of 2D equations. However, instead of two intercepts (x and y), we find three:
  • **X-intercept:** Set \(y = 0\) and \(z = 0\), then solve for \(x\). The solution is where the plane intersects the x-axis.
  • **Y-intercept:** Set \(x = 0\) and \(z = 0\), then solve for \(y\). This is where the plane meets the y-axis.
  • **Z-intercept:** Set \(x = 0\) and \(y = 0\), then solve for \(z\). This shows the intersection on the z-axis.
These intercepts provide specific points that lie on the plane, making them vital references for sketching the full plane. They are especially useful in graphing because a plane in three dimensions can often be perfectly represented by just connecting these three intercepts, forming a triangular slice of the plane.
Graphing planes
Graphing planes in a 3D coordinate system can seem daunting at first, but understanding intercepts makes it more manageable. Once you've found the intercepts:
  • Plot each intercept on the corresponding axis within the 3D space.
  • Draw lines between each set of intercepts. These lines show where the plane touches each axis and helps to create a visual framework of the plane.
  • The result is a triangular outline, which is a part of the plane in the coordinate system.
This basic technique serves as a powerful tool not only for simple educational exercises but in real-world problems, too, where visualizing spatial relationships is crucial.
When you're graphing a plane, the triangle formed by intercepts is part of an infinite plane, extending beyond these initial points. The triangle merely provides a starting glimpse, helpful for understanding the plane's orientation and slope.

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