Problem 7 - Calculus an Applied Approach

Question

Find the intercepts and sketch the graph of the plane. $$ z=8 $$

Step-by-Step Solution

Answer

The z-intercept of the plane is at the point (0, 0, 8). The plane is parallel to the xy-plane and is shifted 8 units upwards along the z-axis. As such, on a graph, it's represented as a flat plane parallel to the xy-plane that intersects the z-axis at z=8.

Step 1: Identify the Axis Intercepts

The given equation is \(z = 8\). It doesn't include \(x\) or \(y\), so there will not be any x or y intercepts. The z-intercept can be found when both \(x\) and \(y\) are set to 0. Here, there is no dependence on \(x\) or \(y\), so the z-intercept is at \(z = 8\). Thus, the z-intercept is at the point (0, 0, 8).

Step 2: Understand the nature of the plane

Since the plane is defined purely in terms of \(z\), it is parallel to the xy-plane (the plane where \(z = 0\)). The only difference is that instead of being at \(z = 0\), it is situated at \(z = 8\). Therefore, it is just a shift of the xy-plane upwards by a distance of 8 units.

Step 3: Sketch the plane

After understanding the nature of the plane, sketch it on a 3D axis. Draw a flat plane parallel to the xy-plane and passing through the z axis at a point representing a value of 8. There's no need to worry about the x and y dimensions, as they extend to infinity on this plane.

Key Concepts

Plane Equations

A plane equation in three-dimensional geometry is a mathematical expression that defines a plane in the 3D coordinate system. One common form of a plane equation is given by \[ ax + by + cz = d \]In this equation, \(a\), \(b\), and \(c\) are constants that define the orientation of the plane, while \(d\) specifies its position relative to the origin. The variables \(x\), \(y\), and \(z\) are the coordinates of any point on the plane.
When a plane is defined, like in the equation \(z = 8\), it indicates a constant height above the \(xy\)-plane, which is zero. Here, this plane does not depend on \(x\) or \(y\), suggesting that it stretches indefinitely along those dimensions.
Understanding plane equations allows us to visualize how they slice through the 3D space, separating it into different regions or layers.

Intercepts

Intercepts in the context of graphing often refer to the points where a graph intersects the axes on a coordinate system.

• **x-intercepts** are where the graph intersects the x-axis. For this to occur, both y and z need to be zero.
• **y-intercepts** occur where the graph intersects the y-axis, requiring x and z to be zero.
• **z-intercepts** are found where the graph intersects the z-axis, achieved by setting x and y to zero. This determines the height of the plane above the xy-plane.

For the plane defined by \(z = 8\), the only intercept is a z-intercept at the point (0, 0, 8). This occurs because there are no x or y terms. Consequently, the plane doesn't intersect the x or y axes, showing how unique this plane appears in the 3D space.

Coordinate Systems

In 3D graphing, coordinate systems are essential for visualizing and understanding spatial relationships. The most commonly used system is the Cartesian coordinate system, consisting of three axes:

• The **x-axis**, typically running horizontally
• The **y-axis**, usually oriented vertically
• The **z-axis**, which adds the depth dimension

Each axis is perpendicular to the others, forming a right-handed system that makes the representation of 3D objects intuitive and straightforward.
When plotting a plane like \(z=8\) in this system, it helps to think about where the plane sits in relation to these axes. In this case, the plane is parallel to the xy-plane but positioned at a height of 8 units along the positive z-axis. This visualization can be thought of as a floating sheet at constant height, simplifying the idea of 3D graphing and the importance of coordinate axes to describe such surfaces.