Problem 40 - Essential Calculus Early Transcendentals

Question

Find an equation of the plane with \(x\) -intercept \(a, y\) -intercept \(b,\) and \(z\) -intercept \(c .\)

Step-by-Step Solution

Answer

The equation of the plane is \(\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1\).

Step 1: Understanding the Plane's Intercepts

The plane intersects the axes at three points: - On the x-axis at point \((a, 0, 0)\)- On the y-axis at point \((0, b, 0)\)- On the z-axis at point \((0, 0, c)\). We need to find an equation of the plane that contains these three points.

Step 2: Using the Intercept Form of the Plane Equation

The equation of a plane with intercepts can be written as \(\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1\). This form directly incorporates the x, y, and z intercepts provided.

Step 3: Final Formula for the Plane

Substitute the intercepts into the plane equation form: \[ \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \]. This is the equation of the plane with x-intercept \(a\), y-intercept \(b\), and z-intercept \(c\).

Key Concepts

X-Intercept

The x-intercept of a plane is the point where the plane crosses the x-axis. To find this intercept for a plane equation, we set the variables corresponding to the y- and z-axes to zero. Let's look into what this means in more detail.

The x-intercept is found when the plane intersects the x-axis at the point \( (a, 0, 0) \). This means that at this point, y and z are both zero. Imagine a plane in three dimensions; the x-intercept is where this plane touches or cuts through the x-axis, indicating that both vertical (y) and depth (z) coordinates are zero.

This concept is crucial, as it is a building block for determining the equation of the plane. When working with the intercept form of the equation of a plane \(\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1\), the x-intercept is represented by \(a\). By substituting zero for y and z in this formula, you can directly find the distance along the x-axis where the plane meets it. Knowing this helps to better visualize the orientation of the plane in a 3D space.

Y-Intercept

The y-intercept of a plane is the point where the plane crosses the y-axis. At this juncture, the plane intersects with the y-axis at point \( (0, b, 0) \). This occurs when both the x and z coordinates are zero; thus, only the y coordinate has a value.

The concept of the y-intercept is pivotal in laying out the geometry of the plane. When we talk about a plane's intercept, the y-intercept shows the location on the y-axis where the plane cuts through the plane of the y-axis and accounts for how far up or down the plane touches the vertical measure.

In the intercept form of a plane's equation \(\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1\), the term \(b\) signifies the y-intercept. By setting x and z to zero, the y-value 'b' shows how elevated the intercept is on the y-axis. This visualization allows us to comprehend how the plane is aligned in three-dimensional space, showing its perpendicular alterations.

Z-Intercept

The z-intercept of a plane is where the plane intersects the z-axis. This particular intercept is located at \( (0, 0, c) \), occurring when both x and y are zero. The z-intercept signals the point at which the plane touches or slices through the z-axis.

Understanding the z-intercept gives insight into a plane's orientation concerning the depth or third dimension. It illustrates where, along the z-axis, the plane exists, which is vital in understanding the spacial dynamics in a three-dimensional context.

In the intercept-form equation \(\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1\), the term \(c\) represents the z-intercept. By substituting zero for x and y, it highlights how far along the z-axis the plane extends. Grasping the role of the z-intercept in the equation helps one form a mental image about the spread and placement of the plane relative to the z-axis, enriching the understanding of its overall orientation in 3D space.