Problem 56 - Precalculus with Calculus Previews

Question

Graph the plane whose equation is given. $$ 3 x+4 y-2 z-12=0 $$

Step-by-Step Solution

Answer

Plot intercepts (4,0,0), (0,3,0), (0,0,-6) and draw plane through them.

Step 1: Identify the Equation Format

The given equation is in the form of a plane equation in three-dimensional space, which is generally represented as \( ax + by + cz + d = 0 \). Here, \( a = 3 \), \( b = 4 \), \( c = -2 \), and \( d = -12 \).

Step 2: Find the Intercepts

To graph the plane, we will find the x-, y-, and z-intercepts. The intercepts are points where the plane crosses the x-, y-, and z-axes, respectively. - **x-intercept**: Set \( y = 0 \) and \( z = 0 \) and solve for \( x \): \( 3x = 12 \) gives \( x = 4 \). The point is \( (4, 0, 0) \).- **y-intercept**: Set \( x = 0 \) and \( z = 0 \) and solve for \( y \): \( 4y = 12 \) gives \( y = 3 \). The point is \( (0, 3, 0) \).- **z-intercept**: Set \( x = 0 \) and \( y = 0 \) and solve for \( z \): \(-2z = 12 \) gives \( z = -6 \). The point is \( (0, 0, -6) \).

Step 3: Plot the Intercepts

On a three-dimensional coordinate system, mark the intercepts: \( (4, 0, 0) \), \( (0, 3, 0) \), and \( (0, 0, -6) \). These points will help you to visualize the plane.

Step 4: Draw the Plane

Using the intercepts plotted in the previous step, draw a plane that passes through these points. Since a plane is defined by at least three non-collinear points, the intercepts help in determining the orientation and position of the plane within the 3D space.

Key Concepts

Intercepts

When visualizing a plane in three-dimensional space, finding the intercepts is a crucial step. Intercepts are essentially where the plane meets the coordinate axes: the x, y, and z axes. Think of them as the footprints of the plane as it touches each axis.
The intercepts serve as reference points that guide us in sketching the plane accurately. To determine each one:

• X-intercept: Set both y and z to zero and solve the equation for x.
• Y-intercept: Set x and z to zero and solve for y.
• Z-intercept: Set x and y to zero and solve for z.

In our specific case, the equation given is: \[3x + 4y - 2z - 12 = 0\]By performing the aforementioned substitutions, you'll find the x-intercept at (4,0,0), the y-intercept at (0,3,0), and the z-intercept at (0,0,-6). These points are fundamental for graphing the plane.

Three-Dimensional Coordinate System

Understanding the three-dimensional coordinate system is fundamental when working with planes. The 3D coordinate system adds depth to the 2D grid you might already be familiar with.
In a 3D coordinate system, each point is determined by three values:

• X-axis: This runs horizontally, just like in a normal 2D graph.
• Y-axis: This also lies in the horizontal plane, but perpendicular to the x-axis.
• Z-axis: This is what makes it three-dimensional, adding vertical depth to the coordinates.

The origin, where all three axes meet, is at (0,0,0). When sketching planes, the 3D coordinate system allows us to define positions and readily visualize objects like planes in space. The coordinate system helps us mark out and connect points, ultimately helping to create a visual impression of complex structures.

Plane Equation

The plane equation is a vital tool in 3D graphing. It's generally expressed as:\[ax + by + cz + d = 0\]Here, \(a\), \(b\), and \(c\) are the coefficients that determine the plane's orientation in space, while \(d\) affects its actual position relative to the origin.
To graph a plane, you need at least three non-collinear points. This guarantees that they are not all in a single line, thus adequately defining the plane's position. In our task, the intercepts we calculated serve as these crucial points.
The given equation, \(3x + 4y - 2z - 12 = 0\), when rearranged into standard form, reveals how changes in these coefficients influence the angle and location of the plane. Understanding the plane equation enables us not only to graph it effectively but also to manipulate variables accordingly to explore different orientations and positions.