Problem 51 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the plane represented by the linear equation. Then list four points that lie in the plane. $$3 x+2 y+2 z=12$$

Step-by-Step Solution

Answer

The plane referred by the equation \(3x + 2y + 2z = 12\) can be first sketched by identifying the points where the plane intersects the axes, then connecting these points to form the plane. Four points that lie on this plane are (4,0,0), (0,6,0), (0,0,6) and (2,2,1).

Step 1: Sketching the Plane

The points where the plane intersects the x, y, and z axes can be found by setting two variables to zero and solving for the remaining variable. Setting y=z=0 gives the point on the x-axis as \(x=12/3=4\), which is (4,0,0). Similarly, setting x=z=0 gives the point on the y-axis as \(y=12/2=6\), which is (0,6,0). Setting x=y=0 gives the point on the z- axis as \(z=12/2=6\), which is (0,0,6). These points are plotted and connected to form a triangle representing the plane.

Step 2: Finding Points on the Plane

Four arbitrary points on the plane can be found by using the original equation \(3x+2y+2z=12\). Three of these points are already found in step 1 - (4,0,0), (0,6,0), and (0,0,6). For the fourth point, we can assume a value for one variable and solve for the others. For instance, let's keep \(x=2\). Then the equation becomes \(6+2y+2z=12\), which simplifies to \(2y+2z=6\), or \(y+z=3\). When \(z=1\), we get \(y=2\). Therefore, the fourth point is (2,2,1).

Key Concepts

Sketching Planes

To sketch a plane represented by a linear equation, we need to visualize it in a three-dimensional coordinate system. A plane can be thought of as a flat sheet that extends infinitely in two dimensions. In our problem, we have the equation \(3x + 2y + 2z = 12\). This equation geometrically represents a plane in 3D space.
To sketch the plane, we first need to identify its intercepts with the coordinate axes:

• Find where it crosses the x-axis by setting \(y\) and \(z\) to zero.
• Find where it crosses the y-axis by setting \(x\) and \(z\) to zero.
• Find where it crosses the z-axis by setting \(x\) and \(y\) to zero.

These intercepts are plotted and connected. This forms a triangular region on the plane, giving us a visual approximation of the plane in space. Remember that even though we sketch a finite piece, it extends infinitely.

Finding Intercepts

Finding intercepts involves determining where the plane intersects the x, y, and z axes. These intercepts give key points on the plane, helping us visualize it in a 3D space. Let's follow the steps:

• X-axis intercept: Set \(y = 0\) and \(z = 0\) in the equation \(3x + 2y + 2z = 12\). Solving \(3x = 12\), we find \(x = 4\). Therefore, the intercept is \((4, 0, 0)\).
• Y-axis intercept: Set \(x = 0\) and \(z = 0\). This gives \(2y = 12\), leading to \(y = 6\). The intercept here is \((0, 6, 0)\).
• Z-axis intercept: Set \(x = 0\) and \(y = 0\). Solving \(2z = 12\), we find \(z = 6\). Thus, the intercept is \((0, 0, 6)\).

These intercepts are crucial when sketching the plane, as they provide the points to draw the triangular representation of the plane in our coordinate system.

Solving Equations for Points

Once you have the basic sketch of the plane, you may want to find additional points on it. This helps confirm your understanding and ensures the plane is sketched correctly. Using our equation \(3x + 2y + 2z = 12\), we can find more points by setting one variable and solving for the others.
Suppose we choose \(x = 2\). Substituting into the equation gives \(6 + 2y + 2z = 12\). Simplify to get \(2y + 2z = 6\), or \(y + z = 3\). This equation represents a line on the plane. You can choose any values for one of these variables and solve for the other. For example, setting \(z = 1\) leads to \(y = 2\), giving us the point \((2, 2, 1)\).
Feel free to try different values. This process not only helps find additional points but offers deeper insight into how linear equations define planes in three dimensions.

3D Coordinate Systems

A 3D coordinate system is an extension of the familiar 2D coordinate system, adding depth with the z-axis. This is vital for understanding and visualizing equations like our plane, \(3x + 2y + 2z = 12\). In 3D space, each point is defined by a triplet \((x, y, z)\), indicating its position along the x, y, and z axes respectively.
To work effectively in a 3D coordinate system:

• Understand how each axis interacts. Each point on a plane like ours has specific x, y, and z values that satisfy the plane equation.
• Visualization is key. Imagine the x-y plane coordinately horizontal, with the z-axis rising vertically.
• Keep in mind, despite sketches being two-dimensional representations, the actual object is three-dimensional.

By mastering these concepts, students can visualize and solve problems related to planes and other geometric figures in three dimensions proficiently.