Problem 59 - Thomas Calculus

Question

Find a formula for the distance from the point \(P(x, y, z)\) to the $$ \text { a. } x \text { -axis. } \quad \text { b. } y \text { -axis. } \quad \text { c. } z $$

Step-by-Step Solution

Answer

a. \(\sqrt{y^2 + z^2}\); b. \(\sqrt{x^2 + z^2}\); c. \(\sqrt{x^2 + y^2}\).

Step 1: Identify the point and axis for the distance calculation

The point given is \(P(x, y, z)\). We need to find a formula for the distance from this point to the \(x\)-axis. On the \(x\)-axis, the points have the form \((x, 0, 0)\).

Step 2: Use the distance formula for point to x-axis

The distance formula between two points \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) is:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]For the point \(P(x, y, z)\) and the nearest point on the \(x\)-axis \((x, 0, 0)\), the formula simplifies to:\[d = \sqrt{(0 - y)^2 + (0 - z)^2} = \sqrt{y^2 + z^2}\]

Step 3: Identify the point and axis for the distance calculation to y-axis

For the distance to the \(y\)-axis, consider that the y-axis has points of form \((0, y, 0)\).

Step 4: Use the distance formula for point to y-axis

Again, apply the distance formula for the point \(P(x, y, z)\) and the nearest point on the \(y\)-axis \((0, y, 0)\). The formula becomes:\[d = \sqrt{(x - 0)^2 + (z - 0)^2} = \sqrt{x^2 + z^2}\]

Step 5: Identify the point and z for the distance calculation to z-axis

To find the distance from the point to the z-axis, the z-axis points are in the form \((0, 0, z)\).

Step 6: Use the distance formula for point to z-axis

Using the distance formula on the point \(P(x, y, z)\) and the nearest point on the \(z\)-axis \((0, 0, z)\), the formula simplifies to:\[d = \sqrt{x^2 + y^2}\]

Key Concepts

3D Coordinate System

Understanding the 3D coordinate system is essential when working with distances between points and axes. In 3D space, each point is located using three coordinates: \(x, y, z\). These coordinates correspond to the point’s position along the x-axis, y-axis, and z-axis respectively.

Visualizing this, imagine a cube where each corner can be defined by different combinations of these three values.

• The x-axis runs horizontally across.
• The y-axis runs vertically up and down.
• The z-axis runs in and out of the flat surface (imagine depth).

This spatial perspective is crucial when determining distances, as each axis provides a different dimension to measure against.

Mastering this spatial reasoning in 3D will help you visualize the problems better and apply formulas accurately.

Point-to-Axis Distance

In 3D geometry, finding the distance from a point to an axis is a common task. Each axis in 3D space represents a line where either the x, y, or z-value is zero.

For context:

• The x-axis is represented by points like (x, 0, 0).
• The y-axis by (0, y, 0).
• The z-axis by (0, 0, z).

The distance between a point and an axis can be thought of as the shortest distance to any point lying directly on that line. This idea applies the concept from 2D to an additional dimension and requires stripping away one of the two non-contributing coordinates.

Calculating these distances is important in scenarios like physics for trajectories, computer graphics to render images, or even in spatial analysis in geosciences.

Distance Calculation Steps

To calculate the distance from a point to an axis in 3D, we follow a specific set of steps using the distance formula derived from the Pythagorean theorem. Here's how this unfolds:

1. **Identify the relevant coordinates:** For each axis that you are calculating the distance from, identify the coordinates on that axis:

• For the x-axis, the coordinates on an axis point will be (x, 0, 0).
• For the y-axis, it will be (0, y, 0).
• For the z-axis, it will be (0, 0, z).

2. **Apply the distance formula:** Use the formula \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\].

Since one coordinate becomes zero, it simplifies:

• From a point \(P(x, y, z)\) to the x-axis: \[d = \sqrt{y^2 + z^2}\].
• To the y-axis: \[d = \sqrt{x^2 + z^2}\].
• To the z-axis: \[d = \sqrt{x^2 + y^2}\].

3. **Interpret the result:** The final result is the shortest path from the point to any point on the specified axis. This shows how distance reduces complexities of dimensions, focusing only on the two that need the calculation.

These steps highlight the systematic approach to breaking down complex spatial problems into manageable calculations.