Problem 27 - APEX Calculus

Question

In Exercises 27-30, find the given distances. The distance from the point (1,2,3) to the plane \(3(x-1)+(y-2)+5(z-2)=0\)

Step-by-Step Solution

Answer

The distance is \(\frac{\sqrt{35}}{7}\).

Step 1: Identify the Problem

We need to find the distance from a point to a plane in space. The point given is \((1, 2, 3)\), and the plane is described by the equation \(3(x-1)+(y-2)+5(z-2)=0\).

Step 2: Simplify the Plane Equation

First, we rewrite the given plane equation in the standard form. Expanding \(3(x-1) + (y-2) + 5(z-2) = 0\) gives us:\[ 3x - 3 + y - 2 + 5z - 10 = 0 \]Combine like terms to obtain the equation:\[ 3x + y + 5z = 15 \]

Step 3: Use the Distance Formula

The formula to find the distance \(D\) from a point \((x_1, y_1, z_1)\) to a plane \(Ax + By + Cz + D = 0\) is given by:\[ D = \frac{|Ax_1 + By_1 + Cz_1 + D|}{\sqrt{A^2 + B^2 + C^2}} \]For our plane, \(A = 3\), \(B = 1\), \(C = 5\), and after rearranging \(15\) to the right, \(D = -15\). The point is \((x_1, y_1, z_1) = (1, 2, 3)\).

Step 4: Plug In the Values

Now, substitute the values into the formula:\[ D = \frac{|3(1) + 1(2) + 5(3) - 15|}{\sqrt{3^2 + 1^2 + 5^2}} \]Simplifying the numerator, we have:\[ |3 + 2 + 15 - 15| = |5| = 5 \]And simplifying the denominator:\[ \sqrt{3^2 + 1^2 + 5^2} = \sqrt{9 + 1 + 25} = \sqrt{35} \]

Step 5: Calculate the Distance

Now calculate the distance:\[ D = \frac{5}{\sqrt{35}} \]To simplify, multiply numerator and denominator by \(\sqrt{35}\):\[ D = \frac{5\sqrt{35}}{35} = \frac{\sqrt{35}}{7} \]

Step 6: Provide the Final Answer

The distance from the point \((1, 2, 3)\) to the plane \(3x + y + 5z = 15\) is \(\frac{\sqrt{35}}{7}\).

Key Concepts

Distance Formula

To find the shortest distance between a point and a plane in 3D space, we use a specific distance formula. This formula helps to determine how "far away" a point is from a flat surface, like a plane, which is crucial in many geometric applications.
The formula is:

• \[ D = \frac{|Ax_1 + By_1 + Cz_1 + D|}{\sqrt{A^2 + B^2 + C^2}} \]
• In this equation,
  • \(A\), \(B\), and \(C\) are the coefficients from the plane's equation \(Ax + By + Cz + D = 0\).
  • \(x_1\), \(y_1\), \(z_1\) represent the coordinates of the point from which we measure the distance.

This formula is used globally because it derives from the vector projection principles, ensuring the shortest path or perpendicular distance between the point and the plane.
It ensures accurate calculations of distances in numerous fields such as physics, engineering, and architecture.

Plane Equation

A plane in 3D geometry represents a flat, level surface that can extend infinitely in two dimensions. The standard form of a plane's equation is:

• \[ Ax + By + Cz + D = 0 \]

This equation tells us all the points \((x, y, z)\) that lie on the plane.
The coefficients \(A\), \(B\), and \(C\) describe the orientation of the plane in space. For example, in our case:

• The equation \(3x + y + 5z = 15\) implies a plane tilted towards the direction influenced by these coefficients in a three-dimensional system.
• The constant \(D\) affects where the plane crosses the axes in the 3D space.

Understanding the equation of a plane is essential for spatial awareness and tackling problems involving intersections, projections, and distances in 3D geometry.

3D Geometry

3D geometry, or three-dimensional geometry, is the field of mathematics dealing with figures that have length, width, and height.
This contrasts with 2D geometry, which only involves length and width.

• In 3D geometry, we define shapes like cubes, spheres, planes, and more, all of which have an added depth dimension compared to their 2D counterparts.
• Coordinate systems, like the cartesian coordinates \((x, y, z)\), play a vital role in describing positions within this space.

Concepts like planes and vectors are much more complex as they operate in an extra dimension. Understanding these aspects is crucial for visualizing and solving real-world problems where depth plays a significant role, such as in computer graphics and engineering fields.

Vector Mathematics

Vector mathematics is a field of mathematics dealing with quantities having both magnitude and direction. Vectors are essential in 3D geometry, as they help describe directions and positions.

• For example, a vector's magnitude might represent a force, while its direction indicates where this force is applied.
• Vectors are usually represented as \((a, b, c)\), showing their direction and magnitude in three-dimensional space.

In terms of planes, vectors can describe normal vectors (perpendicular to the plane).
This concept is fundamental when calculating distances or angles.
Vector operations, like addition, subtraction, and cross-products, offer techniques to solve geometric problems, making them integral in fields like physics and computer science.