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Chapter 8: Problem 8

Find the distance from s3, 7, 25d to each of the following.\(\begin{array}{ll}{\text { (a) The } x y-\text { plane }} & {\text { (b) The } y z \text { -plane }} \\\ {\text { (c) The } x z \text { -plane }} & {\text { (d) The } x \text { -axis }} \\\ {\text { (e) The } y \text { -axis }} & {\text { (f) The } z \text { -axis }}\end{array}\)

Short Answer

Expert verified
25, 3, 7, \(\sqrt{674}\), \(\sqrt{634}\), \(\sqrt{58}\)

Step by step solution

01

Define the point

The given point is \( (3, 7, 25) \). We need to find its distance from various planes and axes.
02

Distance to XY-plane

The XY-plane is defined by \( z = 0 \). The distance from a point \( (x, y, z) \) to the XY-plane is the absolute value of the z-coordinate. Thus, the distance is \( |25| = 25 \).
03

Distance to YZ-plane

The YZ-plane is defined by \( x = 0 \). The distance from the point \( (3, 7, 25) \) to the YZ-plane is the absolute value of the x-coordinate, \( |3| = 3 \).
04

Distance to XZ-plane

The XZ-plane is defined by \( y = 0 \). The distance from the point \( (3, 7, 25) \) to the XZ-plane is the absolute value of the y-coordinate, \( |7| = 7 \).
05

Distance to X-axis

The X-axis is defined by \( y = 0 \) and \( z = 0 \). The distance formula to the X-axis is \( \sqrt{y^2 + z^2} \). Substituting the values, we get \( \sqrt{7^2 + 25^2} = \sqrt{49 + 625} = \sqrt{674} \).
06

Distance to Y-axis

The Y-axis is specified by \( x = 0 \) and \( z = 0 \). Using the distance formula \( \sqrt{x^2 + z^2} \), we substitute the coordinates to get \( \sqrt{3^2 + 25^2} = \sqrt{9 + 625} = \sqrt{634} \).
07

Distance to Z-axis

The Z-axis requires \( x = 0 \) and \( y = 0 \). Using the distance formula \( \sqrt{x^2 + y^2} \), we get \( \sqrt{3^2 + 7^2} = \sqrt{9 + 49} = \sqrt{58} \).

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Key Concepts

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

Distance to Planes
In 3D space, planes are flat surfaces that divide the coordinate system. The most common planes are the XY-plane, YZ-plane, and XZ-plane. Each of these planes is defined by setting one of the three coordinates (x, y, or z) to zero. For instance:
  • The **XY-plane** is represented by setting z = 0.
  • The **YZ-plane** is represented by x = 0.
  • The **XZ-plane** is represented by y = 0.
To find the distance from a point to a plane, we simply take the absolute value of the coordinate that is not part of the plane's definition. For a point
  • To the XY-plane, use the z-coordinate: \(|z|\).
  • To the YZ-plane, use the x-coordinate: \(|x|\).
  • To the XZ-plane, use the y-coordinate: \(|y|\).
Distance to Axes
In a 3D coordinate system, each axis is a line where two coordinates are always zero. The main axes are the x-axis, y-axis, and z-axis:
  • The **x-axis** is where y = 0 and z = 0.
  • The **y-axis** is where x = 0 and z = 0.
  • The **z-axis** is where x = 0 and y = 0.
To determine the distance from a point to any axis, we use the Pythagorean theorem. You must take the square root of the sum of the squares of the two other coordinates (those not zero) related to that axis:
  • To the x-axis: \( \sqrt{y^2 + z^2} \)
  • To the y-axis: \( \sqrt{x^2 + z^2} \)
  • To the z-axis: \( \sqrt{x^2 + y^2} \)
These formulas come from the standard distance formula in a plane modified to exclude the axis coordinates.
3D Coordinate Geometry
In 3D coordinate geometry, any point in space is described with three coordinates: (x, y, z). Think of this as extending the idea of the 2D plane into a third dimension, which adds depth to the width and height. Here, the axes form a three-dimensional grid:
  • The **X-axis** runs horizontally.
  • The **Y-axis** runs vertically.
  • The **Z-axis** introduces depth, going in and out of the plane.
This three-dimensional system allows us to locate points, form shapes, and measure distances between entities such as points, planes, and axes. By using the appropriate formulas and techniques, we can solve complex spatial problems and apply these concepts to real-world scenarios.
Distance Formula
The distance formula is a fundamental tool used in mathematics to compute the distance between two points. In 3D geometry, we use an extended version of the 2D distance formula. For any two points, \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\), the distance \(d\) is:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]This formula is derived from the Pythagorean theorem. It helps to express the length of the segment connecting the two points in three-dimensional space. By understanding and applying this formula, we can address various problems involving distances in space, such as finding the shortest path between locations or measuring distances to axes or planes.

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Most popular questions from this chapter

$$\begin{array}{l}{\text { Find the unit vectors that are parallel to the tangent line to }} \\ {\text { the parabola } y=x^{2} \text { at the point }(2,4) \text { . }}\end{array}$$

Express the solution to the recursion \(\mathbf{n}_{t+1}=A \mathbf{n}_{t}\) in terms of the eigenvectors and eigenvalues of \(A .\) Use DeMoivre's Theorem to simplify the solution if appropriate. \(A=\left[ \begin{array}{rr}{1} & {-a} \\ {a} & {1}\end{array}\right] \quad\) with \(\mathbf{n}_{0}=\left[ \begin{array}{l}{1} \\ {0}\end{array}\right] \quad\) where \(a\) is a real number

Write each system of linear equations in the form \(A \mathbf{x}=\mathbf{b}\) by identifying the matrix \(A\) and the vectors \(\mathbf{x}\) and b. $$\begin{array}{ll}{\text { (a) } 3 x_{1}-x_{2}=9} & {\text { (b) } \quad 2 x_{1}=10} \\ {2 x_{1}+9 x_{2}=-10} & {3 x_{1}-x_{2}=14}\end{array}$$ $$\begin{array}{rlr}{\text { (c) } \quad x_{1}+x_{2}-x_{3}=0} & {\text { (d) } 2 x_{1}-x_{2}+9 x_{3}=12} \\ {2 x_{1}-2 x_{2}+9 x_{3}=5} & {x_{1}-\frac{1}{2} x_{2}+x_{3}=1} \\ {x_{1}+9 x_{3}=1} & {x_{2}=2}\end{array}$$

Suppose that \(\lambda\) is an eigenvalue of \(A .\) Show that \(\lambda^{2}\) is then an eigenvalue of \(A^{2}\) .

\(\begin{array}{c}{\text { For which values of } k \text { does the system of linear equations }} \\ {\text { have zero, one, or an infinite number of solutions? }} \\ {\text { [Note: Not all three possibilities need occur.] }} \\\ {3 x_{1}+x_{2}=2} \\ {k x_{1}+2 x_{2}=4}\end{array}\)
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