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Chapter 9: Problem 63

In Exercises \(63-70,\) label any intercepts and sketch a graph of the plane. $$ 4 x+2 y+6 z=12 $$

Short Answer

Expert verified
The x-intercept of the plane is (3,0,0), the y-intercept is (0,6,0) and the z-intercept is (0,0,2). The graph of the plane is drawn by plotting the intercepts and forming a triangle from the origin to these three points, where the plane contains this triangle.

Step by step solution

01

Finding the x-intercept

Set y=z=0 in the given equation, that is \(4x + 2(0) + 6(0) = 12\), which simplifies to \(4x = 12\). Solve this equation to determine the x-intercept. The solution is \(x = 12/4 = 3\). Thus the x-intercept is (3,0,0).
02

Finding the y-intercept

Similarly, set x=z=0 in the given equation, that is \(4(0) + 2y + 6(0) = 12\), which simplifies to \(2y = 12\). Solve this equation to determine the y-intercept. The solution is \(y = 12/2 = 6\). Thus the y-intercept is (0,6,0).
03

Finding the z-intercept

Finally, set x=y=0 in the given equation to get to \(4(0) + 2(0) + 6z = 12\), which simplifies to \(6z = 12\). Solve this equation to determine the z-intercept. The solution is \(z = 12/6 = 2\). Thus the z-intercept is (0,0,2).
04

Graphing the plane

Having the intercepts, one can proceed to draw a 3D axes system. The points (3,0,0), (0,6,0), and (0,0,2) are plotted and a triangle is formed from the origin to these three points. The plane will be the one that contains this triangle.

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Key Concepts

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

Understanding the X-Intercept in a 3D Cartesian Plane
The x-intercept of a plane in a 3D Cartesian coordinate system is a point where the plane crosses the x-axis. In simpler terms, it鈥檚 where the values of y and z are zero, and only the x-coordinate has a value. To find it, you start by substituting y = 0 and z = 0 into the plane's equation.

For the equation given in the problem, namely \( 4x + 2y + 6z = 12 \), plugging in these zero values simplifies the equation to \( 4x = 12 \).

To solve for x, you divide both sides by 4, giving \( x = 3 \). Therefore, the x-intercept is at the point (3, 0, 0), which tells you the exact position where the plane slices through the x-axis.

Key Points:
  • The x-intercept is found by setting y and z to zero.
  • It represents how far along the x-axis the plane intersects.
Exploring the Y-Intercept in 3D Coordinate Systems
Finding the y-intercept involves locating the point where the plane intersects the y-axis. This means that both x and z have a value of zero at this intercept, and we only solve for y.

Using the same equation, \( 4x + 2y + 6z = 12 \), setting x = 0 and z = 0 reduces it to \( 2y = 12 \). Solving this gives \( y = 6 \), hence the y-intercept is (0, 6, 0).

What this tells us is where exactly along the y-axis does the plane pass. This intercept is particularly useful for visualizing the plane on a 2D slice of the 3D grid.

Key Takeaways:
  • The y-intercept involves setting both x and z to zero.
  • This gives the specific point where the plane crosses the y-axis.
Identifying the Z-Intercept in Three-Dimensional Space
The z-intercept is a crucial point where the plane intersects with the z-axis. Here, just like with the other intercepts, you set the two other coordinates, x and y, to zero, making it possible to solve for z alone.

When you apply this to our equation \( 4x + 2y + 6z = 12 \), you get \( 6z = 12 \) by setting x = 0 and y = 0. By dividing both sides by 6, you find that \( z = 2 \). Hence, the z-intercept is (0, 0, 2).

This point is vital for understanding how the plane lies in 3D space, providing insight into the plane's 'height' above the xy-plane.

Important Points:
  • The z-intercept is derived by making x and y equal to zero.
  • It shows where the plane crosses over the z-axis.

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