Problem 19 - Calculus: Early Transcendentals

Question

Find the distance from (2,2,2) to \(x+y+z=-1\).

Step-by-Step Solution

Answer

Distance is \(3\sqrt{3}\).

Step 1: Identify Key Components

To find the distance from a point to a plane in 3D space, identify the point as \( (x_0, y_0, z_0) = (2, 2, 2) \) and the equation of the plane as \( ax + by + cz + d = 0 \) where \(a=1, b=1, c=1, d=1\).

Step 2: Use Distance Formula

The formula for the distance from a point \( (x_0, y_0, z_0) \) to a plane \( ax + by + cz + d = 0 \) is: \[ d = \frac{|ax_0 + by_0 + cz_0 + d|}{\sqrt{a^2 + b^2 + c^2}} \]

Step 3: Substitute Values into Formula

Substitute \(x_0 = 2, y_0 = 2, z_0 = 2, a = 1, b = 1, c = 1,\) and \(d = 1\) into the formula. Compute the numerator: \( |1(2) + 1(2) + 1(2) + 1| = |9| = 9 \). Then compute the denominator: \( \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3} \).

Step 4: Calculate the Distance

Now compute the distance: \[ d = \frac{9}{\sqrt{3}} = 3\sqrt{3} \] Simplify the expression if needed. Thus, the distance from the point \( (2,2,2) \) to the plane \( x+y+z=-1 \) is \(3\sqrt{3}\).

Key Concepts

3D space

In mathematics, 3D space refers to a three-dimensional environment where every point is defined by three coordinates, usually denoted as \(x, y, z\). These coordinates represent the position along the three axes that are perpendicular to each other:

• The x-axis, which runs horizontally.
• The y-axis, which runs vertically.
• The z-axis, which runs in depth.

Points in this space are often written as an ordered triplet, such as \((x, y, z)\).
Understanding 3D space is crucial because it provides a framework to comprehend and visualize concepts like distance, geometry, or even physics in a more realistic context. In 3D, lines, planes, and various shapes can interact in complex ways that are not possible in two-dimensions.
When dealing with planes in 3D space, think of them as flat surfaces that extend indefinitely. A plane can be defined using a specific equation, which we'll explore further in the equation of a plane section.

Distance formula

Calculating the distance from a point to a plane in 3D space involves using a specific distance formula. This formula helps find how far away a point is from a given plane that extends throughout 3D space.
The distance formula in this scenario is written as: \[ d = \frac{|ax_0 + by_0 + cz_0 + d|}{\sqrt{a^2 + b^2 + c^2}} \] where:

• \(d\) is the distance from the point to the plane,
• \((x_0, y_0, z_0)\) are the coordinates of the point,
• \(a, b, c\) are the coefficients of the plane equation,
• \(d\) is the constant from the plane's equation.

This formula effectively measures how far, in straight-line distance, the point is from the plane. What's important to note is how the numerator determines how the point relates to the plane, while the denominator adjusts based on the orientation and slope of the plane itself. This clarity helps illustrate why the formula might look complex but operates around a straightforward concept—measuring straight-line distance in 3D.

Equation of a plane

The equation of a plane in 3D space is a powerful tool for defining a flat surface stretching infinitely in two dimensions. It's commonly represented in the form:
\[ ax + by + cz + d = 0 \]In this equation:

• \(a, b,\) and \(c\) are the coefficients that define the orientation of the plane,
• \(x, y,\) and \(z\) are variables representing any point on the plane,
• \(d\) is a constant that shifts the plane along its normal vector.

Each coefficient affects how the plane is tilted or positioned in the 3D space, with these numbers effectively defining the vector normal to the plane.
To find the distance from a point to this plane, one needs to plug the point's coordinates into the plane's equation, simplify, and continue using the distance formula. This key equation allows problems like the one in the original exercise to be systematically approached and solved, paving the way to finding precise geometric relationships in three-dimensional environments.