Problem 23 - Precalculus with Calculus Previews

Question

Find the distance from the point (7,-3,-4): (a) to the \(y z\) -plane (b) to the \(x\) -axis.

Step-by-Step Solution

Answer

(a) 7 (b) 5

Step 1: Understanding the yz-plane

The yz-plane consists of all points where the x-coordinate is zero. This means the yz-plane is described by the equation \(x = 0\).

Step 2: Calculate Distance to yz-plane

To find the distance from a point \((x, y, z)\) to the yz-plane, we use the formula \(\text{Distance} = |x|\) because the yz-plane is where \(x = 0\). For the point \((7, -3, -4)\), the distance is \(|7| = 7\).

Step 3: Understanding the x-axis

The x-axis is the line where both the y-coordinate and z-coordinate are zero \((y = 0, z = 0)\). Therefore, the x-axis consists of all points of the form \((x, 0, 0)\).

Step 4: Calculate Distance to x-axis

To find the distance from a point \((x, y, z)\) to the x-axis, we use the formula \(\sqrt{y^2 + z^2}\). For the point \((7, -3, -4)\), the distance is \(\sqrt{(-3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5\).

Key Concepts

yz-plane distance calculation

When dealing with three-dimensional space, understanding how distances are calculated relative to specific planes is essential. The yz-plane is a two-dimensional plane where the x-coordinate is always equal to zero. This means that any point on the yz-plane has the form

• (0, y, z)

To find the distance from a point

• (x, y, z)

to the yz-plane, we consider only the x-coordinate since the yz-plane is defined by

• (x = 0).

The simplest way to think about this is to realize that moving from any point to the yz-plane involves changing the x-coordinate to zero and leaving the y and z coordinates unchanged. Therefore, the formula for calculating the distance from a point to the yz-plane becomes

• Distance \(= |x|\).

For example, given the point (7, -3, -4), the x-coordinate is 7, so the distance to the yz-plane is simply

• |7| = 7.

3D coordinate geometry

3D coordinate geometry is an extension of the two-dimensional system to a three-dimensional space, where points are represented by ordered triples

• (x, y, z).

Each coordinate denotes the position along the x, y, and z axes, respectively. The system is crucial in analyzing spatial relationships and distances in the three-dimensional world.
3D coordinate geometry allows us to define various geometric entities like planes and lines, identifying intersections and distances between these entities. For instance, the Cartesian coordinate system in 3D is equipped with three mutually perpendicular axes:

• The x-axis (horizontal),
• The y-axis (vertical in the plane),
• The z-axis (vertical out of the plane).

Understanding and visualizing this setup help tackle problems involving the measurement of distances and angles between these axes and other geometric figures or points. Utilizing coordinate geometry techniques, like vectors or matrices, can further simplify complex three-dimensional problems.

distance formula in 3D

In three-dimensional space, the distance between two points or a point and a line (or plane) is derived using specific formulas that integrate the x, y, and z coordinates. A critical example of this is the formula determining the distance between two points

• \((x_1, y_1, z_1)\)

and

• \((x_2, y_2, z_2)\)

in 3D space:

• \[ \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \].

This formula is an extension of the Pythagorean theorem into three dimensions, each squared term representing the distance in one directional component.
For exercises like finding the distance from a point to a defined line such as the x-axis, similar techniques apply. The distance to the x-axis involves calculating based purely on the y and z components, as shown by the formula:

• \( \sqrt{y^2 + z^2} \).

Given a point like (7, -3, -4), the distance from this point to the x-axis would be calculated as:

• \( \sqrt{(-3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \).

This direct application of the distance formula exemplifies its importance in physical and applied sciences where three-dimensional analysis is crucial.