Problem 55 - Precalculus with Calculus Previews

Question

Graph the plane whose equation is given. $$ 3 x+z-6=0 $$

Step-by-Step Solution

Answer

The plane passes through (2, 0, 0) and (0, 0, 6) and is parallel to the y-axis.

Step 1: Rewrite the Equation in Standard Form

To graph the plane, we should identify its intercepts. Start by rewriting the equation from its current form \(3x + z - 6 = 0\) into a more familiar form to identify intercepts if necessary: \[3x + 0y + z = 6\] This ensures that it's in the standard plane form \(Ax + By + Cz = D\) where \(y\) is simply absent in this plane.

Step 2: Find Z-Intercept

To find the z-intercept, set both \(x = 0\) and \(y = 0\). Substitute into the equation: \[3(0) + 0 + z = 6\] This simplifies to \(z = 6\). The z-intercept is at the point \((0, 0, 6)\).

Step 3: Find X-Intercept

To find the x-intercept, set both \(y = 0\) and \(z = 0\). Substitute into the equation: \[3x + 0 + 0 = 6\] This simplifies to \(3x = 6\), which means \(x = 2\). The x-intercept is at the point \((2, 0, 0)\).

Step 4: Plot Points and Graph the Plane

With the x-intercept \((2, 0, 0)\) and the z-intercept \((0, 0, 6)\) determined, plot these points on a 3D coordinate system. Since there is no y-term, the plane will be parallel to the y-axis and linearly passing through these intercepts along the x-z plane. This gives a straight line along with the plane we can visualize.

Key Concepts

3D Coordinate System

When graphing in three dimensions, we use a 3D coordinate system. This system extends the familiar 2D coordinate plane (which has x and y axes) by adding a third axis, the z-axis. In this space, every point is represented by three coordinates:

• The x-coordinate measures the point's horizontal distance from the origin (0,0,0).
• The y-coordinate measures the vertical distance from the x-z plane.
• The z-coordinate measures the distance above or below the x-y plane.

The origin in 3D is still the center where all three axes intersect. To visualize this, imagine the z-axis shooting up into the sky from the flat surface of the 2D plane. This addition allows us to explore geometry and equations in a broader sense, such as surfaces and planes rather than just lines and shapes that exist in two dimensions. Understanding how to navigate and plot within this 3D space is crucial for visualizing and solving equations involving planes.

X-Intercept

In three-dimensional graphing, the x-intercept is the point where the graph of an equation crosses the x-axis. To clarify, the x-axis is the horizontal line in a 3D space where both y and z are 0. To determine the x-intercept:

• Set y = 0: This means the point lies in the x-z plane.
• Set z = 0: This means the point exactly lies on the x-axis.

Substitute these values into your equation to find the x-coordinate. For instance, using the equation we have, when y = 0 and z = 0, the equation becomes:\[3x + 0 = 6\]This simplifies to x = 2, giving us the x-intercept at the point (2, 0, 0). This point is significant as it helps define the location and orientation of a plane in conjunction with other intercepts.

Z-Intercept

The z-intercept of an equation in a 3D coordinate system is the point where the graph intersects the z-axis. The z-axis runs perpendicular to both the x and y axes, extending from the plane they create. To find the z-intercept, both x and y are set to 0:

• Set x = 0: This means moving along the line parallel to the z-axis.
• Set y = 0: This keeps us on the x-z plane.

Substitute these into the equation to solve for the value of z. From the example given, when both x and y are 0, the equation simplifies:\[0x + 0 + z = 6\]Thus, z = 6, giving us the z-intercept at the point (0, 0, 6). This point is not only a part of defining the shape of the plane in a 3D space but also essential in plotting and understanding the plane's dimensions and location relative to the origin.