/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 27 Find an equation of the line seg... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 27

Find an equation of the line segment joining the first point to the second point. $$(2,4,8) \text { and }(7,5,3)$$

Short Answer

Expert verified
Question: Find the equation of the line segment joining the points (2, 4, 8) and (7, 5, 3) in 3-dimensional space. Answer: The equation of the line segment joining the points (2, 4, 8) and (7, 5, 3) is given by $$\frac{x - 2}{5} = \frac{y - 4}{1} = \frac{z - 8}{-5}$$.

Step by step solution

01

Finding Direction Ratios

To find the direction ratios, we need to find the difference of the corresponding coordinates of the two given points. Let's denote the given points as A (2, 4, 8) and B (7, 5, 3). The direction ratios can be found using the following formula: Direction Ratio 1 = x2 - x1 (_latx{a = x_2 - x_1}) Direction Ratio 2 = y2 - y1 (_latx{b = y_2 - y_1}) Direction Ratio 3 = z2 - z1 (_latx{c = z_2 - z_1}) Plugging in the values of the coordinates of points A and B, we get: a = 7 - 2, b = 5 - 4, and c = 3 - 8
02

Calculating the Direction Ratios

Now, we will calculate the values of a, b, and c from the previous step. a = 7 - 2 = 5 b = 5 - 4 = 1 c = 3 - 8 = -5 So, the direction ratios are (5, 1, -5).
03

Using the Symmetrical Form of the Line Equation

Now, we will use the symmetrical form of the line equation to find the equation of the line segment. The symmetrical form of the line equation is given by: $$\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}$$ In our case, point A(2, 4, 8) serves as (x1, y1, z1) and the direction ratios are (5, 1, -5). Plugging these values into the equation, we get: $$\frac{x - 2}{5} = \frac{y - 4}{1} = \frac{z - 8}{-5}$$ This is the equation of the line segment joining the points (2, 4, 8) and (7, 5, 3).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Practical formula for \(\mathbf{N}\) Show that the definition of the principal unit normal vector $\mathbf{N}=\frac{d \mathbf{T} / d s}{|d \mathbf{T} / d s|}\( implies the practical formula \)\mathbf{N}=\frac{d \mathbf{T} / d t}{|d \mathbf{T} / d t|} .\( Use the Chain Rule and Note that \)|\mathbf{v}|=d s / d t>0.$

An object moves along a path given by \(\mathbf{r}(t)=\langle a \cos t+b \sin t, c \cos t+d \sin t\rangle, \quad\) for \(0 \leq t \leq 2 \pi\) a. What conditions on \(a, b, c,\) and \(d\) guarantee that the path is a circle? b. What conditions on \(a, b, c,\) and \(d\) guarantee that the path is an ellipse?

Trajectory with a sloped landing Assume an object is launched from the origin with an initial speed \(\left|\mathbf{v}_{0}\right|\) at an angle \(\alpha\) to the horizontal, where \(0 < \alpha < \frac{\pi}{2}\) a. Find the time of flight, range, and maximum height (relative to the launch point) of the trajectory if the ground slopes downward at a constant angle of \(\theta\) from the launch site, where \(0 < \theta < \frac{\pi}{2}\) b. Find the time of flight, range, and maximum height of the trajectory if the ground slopes upward at a constant angle of \(\theta\) from the launch site.

Show that the two-dimensional trajectory $$x(t)=u_{0} t+x_{0}\( and \)y(t)=-\frac{g t^{2}}{2}+v_{0} t+y_{0},\( for \)0 \leq t \leq T$$ of an object moving in a gravitational field is a segment of a parabola for some value of \(T>0 .\) Find \(T\) such that \(y(T)=0\)

Consider the curve \(\mathbf{r}(t)=(\cos t, \sin t, c \sin t),\) for \(0 \leq t \leq 2 \pi,\) where \(c\) is a real number. It can be shown that the curve lies in a plane. Prove that the curve is an ellipse in that plane.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.