Problem 57 - Precalculus with Calculus Previews

Question

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

Step-by-Step Solution

Answer

The plane intersects the axes at (-4, 0, 0), (0, 2, 0), and (0, 0, 4).

Step 1: Identify the Plane Equation

The equation provided is \(-x + 2y + z = 4\). This equation represents a plane in three-dimensional space.

Step 2: Find the Intercepts

Set two variables to zero to find the intercepts. - For the x-intercept, set \(y = 0\) and \(z = 0\): \(-x = 4\), so \(x = -4\).- For the y-intercept, set \(x = 0\) and \(z = 0\): \(2y = 4\), so \(y = 2\).- For the z-intercept, set \(x = 0\) and \(y = 0\): \(z = 4\), so \(z = 4\).

Step 3: Plot the Intercepts

Plot the points \((-4, 0, 0)\), \((0, 2, 0)\), and \((0, 0, 4)\) on a three-dimensional coordinate system. These points represent where the plane intersects the x, y, and z axes respectively.

Step 4: Draw the Plane

Connect the three intercept points in the coordinate system. Form a triangle among these points as they serve as a guide to draw the plane. The plane extends infinitely in all directions beyond this triangle following the direction of the axis, maintaining the equation \(-x + 2y + z = 4\).

Key Concepts

Plane Equation

The equation \(-x + 2y + z = 4\) represents a plane in a three-dimensional space, where \(x\), \(y\), and \(z\) are the variables corresponding to the X, Y, and Z axes, respectively. A plane equation in this general form is expressed as \(Ax + By + Cz = D\), where \(A\), \(B\), and \(C\) are coefficients influencing the slope and orientation of the plane, and \(D\) is a constant that shifts the plane in space. Here in our example, \(A = -1\), \(B = 2\), \(C = 1\), and \(D = 4\).

Such an equation is essential for understanding how planes interact with three-dimensional spaces. The values of \(A\), \(B\), and \(C\) indicate the steepness and direction of the normal vector to the plane. This vector is perpendicular to the plane and helps determine its orientation. Visualizing a plane requires finding key points where it intersects the coordinate axes, which can be established by solving for intercepts at points where two variables are set to zero. Understanding plane equations is crucial for fields from computer graphics to engineering marvels.

3D Coordinate System

A three-dimensional coordinate system is a framework that uses three axes—commonly labeled as the X, Y, and Z axes—to establish a spatial grid where points can be precisely plotted. Each axis is perpendicular to the other two, providing a way to navigate in three-dimensional space. When plotting a plane like \(-x + 2y + z = 4\), this space allows us to visualize the location of the plane and its interactions with each axis.

Key characteristics of the 3D coordinate system include:

• The origin point, \((0, 0, 0)\), which is the intersection of the three axes.
• Movements along each axis independently affect the point's coordinates, height, width, and depth.
• By using this system, geometric transformations and spatial positioning become feasible, enabling advanced visualization and mathematical operations.

Understanding this coordinate system is vital for interpreting graphs, situating objects in space, and solving complex geometrical problems. Whether it is used in physics to find the trajectory of a projectile, in architecture to balance a structure, or in virtual simulations, mastery of the 3D coordinate system opens doors to comprehensive spatial analysis.

Intercepts

Intercepts are pivotal in determining where a plane crosses the X, Y, and Z axes, providing tangible points for plotting the plane on a graph. Identifying intercepts is a common first step in graphing a plane. To find intercepts in the equation \(-x + 2y + z = 4\), you'd set two variables to zero and solve for the third:

- **X-intercept:** Set \(y = 0\) and \(z = 0\), giving \(-x = 4\). Hence, \(x = -4\) resulting in the point \((-4, 0, 0)\).- **Y-intercept:** Set \(x = 0\) and \(z = 0\), resulting in \(2y = 4\), and thus \(y = 2\) or the point \((0, 2, 0)\).- **Z-intercept:** Set \(x = 0\) and \(y = 0\), leading to \(z = 4\), or the point \((0, 0, 4)\).

Once intercepts are determined, plotting these points on a 3D plane can guide sketching.. These points are connected to form a triangular region, which then aids in visualizing the infinite boundary of the plane as it extends in space beyond the confines of the axes. By focusing on intercepts, students gain a straightforward method to begin graphing and understanding planes, bridging abstract equations with tangible visuals.