Problem 8 - Calculus an Applied Approach

Question

Find the intercepts and sketch the graph of the plane. $$ x=5 $$

Step-by-Step Solution

Answer

The plane given by the equation \(x=5\) intersects the x-axis at the point (5,0,0). The plane is parallel to yz-plane.

Step 1: Identify the intercept

The equation of the given plane is \(x=5\). This means that the plane intersects the x-axis at the point (5,0,0).

Step 2: Identify additional points

The plane is parallel to the yz-plane, that means it holds any point where x=5. This includes all points on the form (5,y,z), for example (5,1,1), (5,2,0), (5,-2,-1) etc.

Step 3: Graph the plane

Draw the axes on the graph. Then plot the points from step 1 and step 2. The plane should be parallel to the yz-plane and intersect the x-axis at the point (5,0,0). As the points (5,1,1), (5,2,0), (5,-2,-1) are on the plane, you should be able to draw a plane that includes these points. Remember, this is a three-dimensional graph, so your sketch should reflect that.

Key Concepts

Understanding Intercepts in Three-Dimensional Geometry

An intercept in three-dimensional geometry refers to the points where a plane intersects the coordinate axes. In three dimensions, we have an x-axis, a y-axis, and a z-axis. Each intercept tells us where the plane touches one of these axes:

• The **x-intercept** is at the point where the plane meets the x-axis.
• The **y-intercept** is where the plane meets the y-axis.
• The **z-intercept** happens at the intersection with the z-axis.

For the exercise given, the equation of the plane is simple: \(x=5\). This tells us that the plane only intersects the x-axis and not the y or z since there are no y or z terms. Therefore, the x-intercept is at (5,0,0), implying the plane crosses the x-axis at x = 5. There are no y or z-intercepts because the plane never touches those axes when x is held constant at 5.

Exploring Planes in 3D Space

Planes in 3D geometry are flat, two-dimensional surfaces that extend infinitely in three-dimensional space. Each plane can be described by a unique equation representing points lying on it. To describe a simple plane like \(x=5\):

• The plane is **parallel** to the yz-plane.
• It contains all points where the x-coordinate is constantly 5, for example, (5,y,z).
• Although it extends infinitely in the y and z directions, it is fixed in the x dimension at x=5.

Visualizing such a plane involves imagining a vertical sheet or wall positioned at x=5 and stretching infinitely along y and z directions, but never moving away from x=5. This makes understanding planes crucial as they help interpret complex geometric relationships in three-dimensional space.

Connecting with Coordinate Geometry

Coordinate geometry, or analytic geometry, is a mathematical discipline that uses a coordinate system to graph and analyze geometrical shapes. In 3D geometry, understanding planes is fundamental:

• Coordinate systems help locate positions by using ordered triples (x,y,z).
• These coordinates allow us to describe locations and equations of planes like \(x=5\).
• By using coordinate geometry, we can plot and visualize how different geometric figures interact and how their positions change relative to the axes.

In our exercise, plotting the intercepts and additional points onto a coordinate system helps us see how the plane behaves in 3D. It demonstrates how a plane's position can be summarized by a simplified equation like \(x=5\), allowing us to predict its behavior and interaction with the axes efficiently and effectively.