Problem 6 - Calculus an Applied Approach

Question

Find the intercepts and sketch the graph of the plane. $$ 2 x-y+z=4 $$

Step-by-Step Solution

Answer

The x-intercept is (2,0,0), y-intercept is (0,-4,0) and z-intercept is (0,0,4).

Step 1: Find the x-intercept

Set y = 0 and z = 0. The equation then reduces to \(2x - 0 + 0 = 4\), upon solving which we get x = 2.

Step 2: Find the y-intercept

Setting x = 0 and z = 0 gives us \(-y + 0 = 4\), which after solving gives y = -4.

Step 3: Find the z-intercept

Setting x = 0 and y = 0 in the equation gives us \(z = 4\).

Step 4: Plotting the Intercepts and Drawing the Plane

Using the x-intercept (2,0,0), the y-intercept (0,-4,0) and the z-intercept (0,0,4), one can sketch the plane by drawing a triangle with vertices at these intercepts. Connect the points and extend them to retain the general form of the plane.

Key Concepts

Intercepts in 3D Geometry

Intercepts are key points where a plane intersects the axes in a 3D coordinate system. They give essential clues about the position and orientation of the plane. To find the intercepts:

• X-intercept: This is found by setting the other two coordinates (y and z) to zero and solving for x. For example, in the equation \(2x - y + z = 4\), set y = 0 and z = 0, resulting in the x-intercept \( (2, 0, 0) \).
• Y-intercept: Set x = 0 and z = 0 to solve for y, giving the point \( (0, -4, 0) \).
• Z-intercept: Set x = 0 and y = 0 to solve for z, resulting in the z-intercept \( (0, 0, 4) \).

These intercepts allow us to sketch the plane in 3D space, providing insight into its spatial orientation and dimensions.
Understanding intercepts helps students tackle complex geometry problems by introducing a systematic approach to graphing and analyzing 3D spaces.

Graphing Planes in 3D

Graphing a plane in 3D involves using intercepts to visualize its position relative to the coordinate axes. Planes in 3D are flat surfaces extending infinitely. However, to graph them, we often represent them in a limited region.
Begin by locating the intercepts calculated in the earlier steps. In this exercise, these are:

• X-intercept at \( (2, 0, 0) \)
• Y-intercept at \( (0, -4, 0) \)
• Z-intercept at \( (0, 0, 4) \)

Next, connect these points to form a triangle. This triangle is a cross-section of the plane in the visible area of the graph. Since a plane is infinite, consider extending the lines beyond the triangle to reflect its larger span. This method provides a simplified representation that aids in understanding the plane's behavior and characteristics in space, fostering a deeper comprehension of spatial relationships.

Understanding Geometry in 3D Coordinates

Geometry in 3D coordinates can initially seem challenging, but it provides a powerful perspective for solving spatial problems. The equations represent different geometric constructs based on their coefficients and constants.
Each term in a plane equation like \(2x - y + z = 4\) contributes to the plane's tilt and position:

• The coefficient of x (2) affects the plane's slope along the x-axis.
• The coefficient of y (-1) influences the plane's orientation along the y-axis.
• The coefficient of z (1) shapes how the plane interacts with the z-axis.

The constant (4) shifts the plane, affecting its distance from the origin.
By understanding these components, students can predict how altering elements of the equation changes the plane's orientation and position in 3D space. Mastery of these concepts provides a foundation for advanced studies in physics, engineering, and computer graphics, where 3D geometry plays a crucial role.