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Chapter 12: Problem 65

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

Short Answer

Expert verified
The distances are 鈭�(y虏 + z虏), 鈭�(x虏 + z虏), and 鈭�(x虏 + y虏) from the x, y, and z axes respectively.

Step by step solution

01

Understand the Distance Formula

The distance in a 3D coordinate system between any point \( (x_1, y_1, z_1) \) and another point \( (x_2, y_2, z_2) \) is given by \( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}.\)
02

Find Distance from Point to the x-axis

The x-axis is composed of points of the form \((x, 0, 0)\). Using the distance formula, the distance from a point \(P(x, y, z)\) to the x-axis is \( \sqrt{(x - x)^2 + (y - 0)^2 + (z - 0)^2} = \sqrt{y^2 + z^2}.\)
03

Find Distance from Point to the y-axis

The y-axis consists of points of the form \((0, y, 0)\). Thus, the distance from the point \(P(x, y, z)\) to the y-axis can be calculated as \( \sqrt{(x - 0)^2 + (y - y)^2 + (z - 0)^2} = \sqrt{x^2 + z^2}.\)
04

Find Distance from Point to the z-axis

The z-axis is defined by points of the form \((0, 0, z)\). Hence, the distance from a point \((P(x, y, z)\) to the z-axis is given by \( \sqrt{(x - 0)^2 + (y - 0)^2 + (z - z)^2} = \sqrt{x^2 + y^2}.\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding x-axis Distance
When dealing with 3D coordinates, distance to the x-axis focuses on the other two dimensions, y and z. Imagine a straight line running through your location but only along the x-axis.
This line excludes any movement in the y or z direction. Therefore, to find the distance from a point \( P(x, y, z) \) to the x-axis, we measure how far the point extends into the y and z coordinates.
Using the 3D distance formula simplifies this task. The formula for distance from a point to the x-axis is \( \sqrt{y^2 + z^2} \).
  • Replace x with itself, leading to no change along x \((x-x=0)\).
  • Calculate the square roots of the squared y and z values.
Essentially, isolate the y and z distances because those are directions orthogonal to the x-axis.
So, when visualizing, envision a point projecting perpendicularly to the x-axis from the yz plane.
Understanding y-axis Distance
Similar to the x-axis scenario, the distance to the y-axis analyzes the domain of space ignoring movement along the y direction. The y-axis is a line where x and z are fixed at zero, and only y varies.
To determine how far a point \(P(x, y, z)\) is from the y-axis, measure the distance based on its x and z projections.
This results in the formula \( \sqrt{x^2 + z^2} \).
  • Since you're looking for deviations from being on the y-axis, ignore changes in the y value \((y-y=0)\).
  • Calculate the square roots of the squared x and z values.
This rightly summarizes any deviation in the x-z plane.
Consider a shadow of the point that's perpendicularly cast onto the xy plane which shows the perception of distance from the y-line.
Understanding z-axis Distance
In 3D coordinates, understanding distance to the z-axis focuses on taking out the z component. Imagine a line extending straight up through your position, ignoring how far you go in the x or y direction.
The z-axis is a vertical line, where only the z-coordinate changes, and x and y remain zero.
To calculate how far a point \(P(x, y, z)\) deviates from the z-axis, consider the movements in x and y only. This results in the formula \(\sqrt{x^2 + y^2}\).
  • You eliminate z's impact since you're measuring perpendicular to the z-line \((z-z=0)\).
  • Compute the square root of the summed squares of x and y.
Your mind's eye should picture a drop or rise in a plane parallel to the xy plane.
This drop is the projection of point P when cast perpendicularly on top of the xy plane.

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Most popular questions from this chapter

Find the center \(C\) and the radius \(a\) for the spheres in Exercises \(51-60\) $$x^{2}+\left(y+\frac{1}{3}\right)^{2}+\left(z-\frac{1}{3}\right)^{2}=\frac{16}{9}$$

Find the center \(C\) and the radius \(a\) for the spheres in Exercises \(51-60\) $$x^{2}+y^{2}+z^{2}+4 x-4 z=0$$

In Exercises \(1-16,\) give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. $$x^{2}+(y-1)^{2}+z^{2}=4, \quad y=0$$

Find the areas of the parallelograms whose vertices are given in Exercises \(35 - 40 .\) $$ A ( 0,0 ) , \quad B ( 7,3 ) , \quad C ( 9,8 ) , \quad D ( 2,5 ) $$

\begin{equation} \begin{array}{l}{\text { In Exercises } 35-44 \text { , describe the given set with a single equation or }} \\ {\text { with a pair of equations. }}\end{array} \end{equation} The set of points in space that lie 2 units from the point \((0,0,1)\) and, at the same time, 2 units from the point \((0,0,-1)\)
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